# Transport equations and flows with one-sided Lipschitz velocity fields

Pierre-Louis Lions  Université Paris-Dauphine and Collège de France
<EMAIL_ADDRESS>Benjamin Seeger  University of Texas at Austin
<EMAIL_ADDRESS>
Partially supported by the National Science Foundation under award number
DMS-1840314

###### Abstract

We study first- and second-order linear transport equations, as well as ODE
and SDE flows, with velocity fields satisfying a one-sided Lipschitz
condition. Depending on the time direction, the flows are either compressive
or expansive. In the compressive regime, we characterize the stable continuous
distributional solutions of both the first and second-order nonconservative
transport equations as the unique viscosity solution. Our results in the
expansive regime complement the theory of Bouchut, James, and Mancini [23],
and we provide a complete theory for both the conservative and nonconservative
equations in Lebesgue spaces, as well as proving the existence, uniqueness,
and stability of the regular Lagrangian ODE flow. We also provide analogous
results in this context for second order equations and SDEs with degenerate
noise coefficients that are constant in the spatial variable.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
      1. 1.1.1 The compressive regime
      2. 1.1.2 The expansive regime
      3. 1.1.3 SDEs and second order equations
    2. 1.2 Applications and further study
    3. 1.3 Notation
  2. 2 The ODE flow
    1. 2.1 The backward flow
    2. 2.2 The Jacobian for the backward flow
    3. 2.3 The forward flow as the right-inverse of the backward flow
    4. 2.4 Compressive stochastic flows
    5. 2.5 Small noise approximations
  3. 3 The compressive regime
    1. 3.1 The nonconservative equation
      1. 3.1.1 Representation formula
      2. 3.1.2 Viscosity solutions
      3. 3.1.3 (Non)equivalence of distributional and viscosity solutions
    2. 3.2 The conservative equation
      1. 3.2.1 Duality solutions
      2. 3.2.2 On the failure of renormalization
      3. 3.2.3 Equivalence of duality and distributional solutions
  4. 4 The expansive regime
    1. 4.1 The conservative equation
      1. 4.1.1 Representation formula
      2. 4.1.2 Vanishing viscosity approximation
    2. 4.2 The nonconservative equation
      1. 4.2.1 $L^{p}$ and $BV$ estimates
      2. 4.2.2 Duality solutions
      3. 4.2.3 Renormalization
    3. 4.3 The forward ODE flow
      1. 4.3.1 Properties of the right inverse
      2. 4.3.2 The regular Lagrange property
    4. 4.4 Characterizations
      1. 4.4.1 The nonconservative equation: $\sup$ and $\inf$ convolutions
      2. 4.4.2 The conservative equation: uniqueness of nonnegative solutions
      3. 4.4.3 Uniqueness of regular Lagrangian flows
    5. 4.5 Some remarks for second order equations
      1. 4.5.1 The expansive stochastic flow with constant noise coefficient
      2. 4.5.2 A priori estimates for the second-order nonconservative equation
      3. 4.5.3 Representation formula for the Fokker-Planck equation

##  1 Introduction

For a fixed, finite time horizon $T>0$ and a velocity field
$b:[0,T]\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$, we study the linear
transport equation

(1.1) $\partial_{t}u+b(t,x)\cdot\nabla u=0\quad\text{in
}[0,T]\times{\mathbb{R}}^{d},$

along with the dual, continuity equation

(1.2) $\partial_{t}f+\operatorname{div}(b(t,x)f)=0\quad\text{in
}[0,T]\times{\mathbb{R}}^{d},$

and the associated ordinary differential equation (ODE) flow

(1.3)
$\partial_{t}\phi_{t,s}(x)=b(t,\phi_{t,s}(x)),\quad(s,t,x)\in[0,T]\times[0,T]\times{\mathbb{R}}^{d},\quad\phi_{s,s}=\operatorname{Id}.$

The goal of the paper is to analyze the three problems, and the relations
between them, for vector fields $b$ satisfying the one-sided Lipschitz
condition

(1.4)
$\left\\{\begin{split}&(b(t,x)-b(t,y))\cdot(x-y)\geq-C(t)|x-y|^{2}\quad\text{for
a.e. }(t,x,y)\in[0,T]\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\\\
&\text{for some nonnegative $C\in L^{1}([0,T])$}.\end{split}\right.$

When $b$ is Lipschitz continuous in the space variable, the ODE flow (1.3)
admits a unique global solution, and, through the method of characteristics,
(1.1) and (1.2) are uniquely solved for any given smooth initial or terminal
data. Moreover, the flow is a diffeomorphism, and therefore the solution
operators for either the initial value problem (IVP) or terminal value problem
(TVP) for (1.1) and (1.2) are continuous on $L^{p}_{\mathrm{loc}}$ for any
$p\in[1,\infty]$.

Under the assumption (1.4), the time direction plays a nontrivial role, and
there is a fundamental difference between the solvability of the flow (1.3)
forward versus backward in time. Indeed, $b$ need not even be continuous, and
(1.4) is equivalent to

$\frac{\nabla b(t,\cdot)+\nabla
b(t,\cdot)^{T}}{2}\geq-C(t)\operatorname{Id}\quad\text{in the sense of
distributions.}$

In particular, the distribution $\operatorname{div}b$ is a signed measure that
is bounded from below, but not in general absolutely continuous with respect
to Lebesgue measure. Thus, when $t<s$, the flow (1.3) is expected to
concentrate at sets of Lebesgue measure zero, while the formation of vacuum is
witnessed for $t>s$.

A general study of transport equations and ODEs with irregular velocity fields
was initiated by DiPerna and the first author [37], who introduced the notion
of renormalized solutions to prove the well-posedness for (1.1) and (1.2) and
the almost-everywhere solvability of the flow (1.3) for $b$ with Sobolev
regularity. The DiPerna-Lions theory was extended to equations where only
$\mathrm{Sym}(\nabla b)\in L^{1}$, to Vlasov equations with
$BV_{\mathrm{loc}}$ velocity fields [18], and to two-dimensional problems with
a Hamiltonian structure [1, 2, 3, 19, 42]. Using deep results from geometric
measure theory, the renormalization property was extended to the very general
case where $b\in BV_{\mathrm{loc}}$ and $\operatorname{div}b\in L^{1}$ by
Ambrosio [4], who also provided a new, measure-theoretic viewpoint on the
relationship between uniqueness of nonnegative solutions of (1.2) and the
unique solvability of the flow (1.3) through the idea of superposition.
Further developments include equations with velocity fields having a
particular structure allowing for less regularity [47, 28] and velocity fields
belonging to $SBD$ (i.e. $\mathrm{Sym}(\nabla b)$ is a signed measure with no
singular Cantor-like part) [12]. Fine regularity properties of DiPerna-Lions
flows were established in [15, 33], and the study of so-called “nearly
incompressible flows” [14] led to the resolution by Bianchini and Bonicatto
[17] of Bressan’s compactness conjecture [25, 24]; see also [44] for related
results. For many more details and references, we refer the reader to the
surveys [5, 6, 16, 11].

In the majority of these works, the divergence $\operatorname{div}b$ is
assumed to be bounded, or at least absolutely continuous with respect to
Lebesgue measure. This is not the case in general for velocity fields
satisfying (1.4), and so the equations (1.1) and (1.2) do not even have a
sense as distributions, because the products $(\operatorname{div}b)u$ and $bf$
are ill-defined for general $u\in L^{1}_{\mathrm{loc}}$ or measures $f$. The
DiPerna-Lions theory does not, therefore, cover this situation. Moreover, the
choice of an appropriate function space of solutions is very sensitive to
whether the equations are posed as initial or terminal value problems.

The problems (1.1)-(1.3) for velocity fields with a one-sided Lipschitz
condition have been approached with a variety of methods [27, 55, 56, 30, 57,
20, 23]. Our main purpose is to complement these works, and in particular the
theory of Bouchut, James, and Mancini [23], by providing complete
characterizations of the stable solutions to all three problems in both the
compressive and expansive regimes. We also provide some results on the
corresponding parabolic equations with a degenerate, second-order term, as
well as the SDE analogue of (1.3) for both the velocity field $b$ and $-b$.

###  1.1 Main results

We relegate a full description of the results, discussions, and examples to
the body of the paper. Here, we briefly outline the different sections and the
types of results proved within them, and we compare them to the existing
literature.

####  1.1.1 The compressive regime

In Section 2, we record properties of the backward Filippov flow for (1.3), as
well as for its Jacobian $J_{t,s}(x):=\det(\nabla\phi_{t,s}(x))$, which is
well-defined in $L^{\infty}$ for a.e. $t\leq s$ and $x\in{\mathbb{R}}^{d}$. We
employ measure-theoretic arguments to make sense of the right-inverse of the
flow in an almost-everywhere sense, as a preliminary step to understanding the
forward, regular Lagrangian flow, and prove several properties, the most
important of which is its almost-everywhere continuity.

In Section 3, we turn to the study of the nonconservative equation111For a
consistent presentation throughout the paper, and in order to emphasize the
dual relationship between the two equations, the transport equation (1.1) will
always be posed as a terminal value problem, and the continuity equation (1.2)
as an initial value problem. The compressive and expansive regimes will be
distinguished by the choice of sign in front of the velocity field $b$.

(1.5) $\partial_{t}u-b(t,x)\cdot\nabla u=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad u(T,\cdot)=u_{T},$

for which the uniqueness of continuous distributional solutions fails in
general. We introduce a new PDE characterization of the “good” (stable)
solution of (1.5) as the unique viscosity solution, in the sense of Crandall,
Ishii, and the first author [32]. This is done by proving a comparison
principle for sub and supersolutions. The viscosity solution characterization
coincides with the selection of “good” solutions by other authors in
particular settings [30, 57, 55, 56, 20, 23], allows for robust stability
statements, and, moreover, generalizes to the setting of degenerate parabolic
problems (see the discussion below).

The “usual” viscosity solution theory must be modified due to the lack of
global continuity of $b$. In view of the evolution nature of the equations,
the $L^{1}$-dependence in time does not present a problem, and the equations
can be treated with the methods of [53, 54, 43, 51]. To deal with the
discontinuity of $b$ in space, sub and supersolutions must be defined with
appropriate semicontinuous envelopes of $b$ in the space variable. The
direction of the one-sided Lipschitz assumption (1.4) accounts for the
beneficial inequalities in the proof of the comparison principle.

We then introduce further conditions on the velocity field $b$ and terminal
data $u_{T}$ that ensure uniqueness of arbitrary continuous distributional
solutions. In particular, the interplay between the regularity of $b$ and
$u_{T}$ plays an important role: if $b\in C^{\alpha}$ and $u_{T}\in
C^{\beta}$, then distributional solutions are unique if $\alpha+\beta>1$,
while uniqueness may fail in general if $\alpha+\beta\leq 1$, as we show by
example.

The latter half of Section 3 deals with the study of the dual problem to
(1.5), namely

(1.6) $\partial_{t}f-\operatorname{div}(b(t,x)f)=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}\quad f(0,\cdot)=f_{0}.$

Even if $f_{0}\in L^{1}_{\mathrm{loc}}$, the concentrative nature of the flow
causes the measure $f(t,\cdot)$ to develop a singular part, and therefore we
are led to seek measure-valued solutions. This prevents the duality solution
of (1.6) from being understood in the distributional sense, due to the lack of
continuity of $b$. Nevertheless, we prove that, if $b$ is continuous, or if it
happens that $f(t,\cdot)$ is absolutely continuous with respect to Lebesgue
measure on the time interval $[0,T]$, then the notions of duality and
distributional solutions are equivalent.

An important feature of the continuity equation (1.6) is the failure of
renormalization; that is, if $f$ is a duality solution, the measure $|f|$ may
fail to be a distributional solution, and may even violate conservation of
mass. This is in contrast with the DiPerna-Lions theory, and is a direct
consequence of the compressive nature of the backward flow, which can lead to
cancellation of the positive and negative parts of $f$. A related phenomenon
is the nonuniqueness of distributional solutions of the continuity equation
(1.2) with the reverse sign (see below).

####  1.1.2 The expansive regime

In Section 4, we reverse the sign on the velocity field, and study the
corresponding problems

(1.7) $\partial_{t}u+b(t,x)\cdot\nabla u=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad u(T,\cdot)=u_{T}$

and

(1.8) $\partial_{t}f+\operatorname{div}(b(t,x)f)=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad f(0,\cdot)=f_{0}.$

In view of the lower bound on the divergence of $b$, we are motivated to seek
an $L^{p}$-based theory for both equations, based on a priori estimates, or
equivalently, on the fact that the characteristic flow (the forward ODE (1.3))
does not concentrate on sets of measure zero.

The initial value problem for the continuity equation (1.8) was studied in
[20, 23], where a large part of the analysis is based on the fact that locally
integrable distributional solutions are _not_ unique in general222Note that in
[23], the velocity field $a$ takes the role of $-b$ in (1.4), while the time-
directions of (1.7) and (1.8) are switched, so their setting corresponds to
the expansive regime discussed here.. The same setting is studied in [27],
where the existence and uniqueness of the forward Filippov flow for (1.3) is
established for a.e. $x\in{\mathbb{R}}^{d}$.

In the first part of Section 4, we identify a unique “good” distributional
solution, and prove that the resulting solution operator is continuous on
$L^{p}_{\mathrm{loc}}$ for all $p\in[1,\infty]$, and stable with respect to
regularizations. This coincides with the notion of reversible solution in [20,
23].

We then obtain strong stability results for the Bouchut-James-Mancini duality
solutions of the nonconservative problem (1.7) in all $L^{p}$-spaces, which
allow us to prove the renormalization property. Moreover, we introduce a PDE
characterization of this duality solution in terms of regularization by
$\operatorname*{ess\,inf}$\- and $\operatorname*{ess\,sup}$-convolution. An
important ingredient in establishing this characterization is the propagation
of almost-everywhere continuity, which, in turn, follows from the
renormalization property and the almost-everywhere continuity of the forward
flow proved in Section 2.

As a consequence of this new characterization, we give a PDE-based proof of
the fact that _nonnegative_ distributional $L^{p}$-solutions of (1.8) are
unique, which was established in [27] using the superposition principle. This
result, along with the renormalization property for (1.7), allows us to
establish the existence, uniqueness, and stability of the forward regular
Lagrangian flow for the ODE (1.3) identified in [27]. As a byproduct, this
also provides a full characterization of the Bouchut-James-Mancini notion of
“good” (reversible) solution as the pushforward of $f_{0}$ by the forward
flow. Moreover, a distributional solution $f$ is a reversible solution if and
only if $|f|$ is also a distributional solution (cf. [23, Proposition 3.12],
which operates under the criterion that $f$ be a so-called “Jacobian”
solution).

####  1.1.3 SDEs and second order equations

This paper also contains various results regarding second order versions of
(1.1) and (1.2), as well as stochastic differential equation (SDE) flows. SDEs
and degenerate second-order Fokker-Planck equations have been studied from
many perspectives, using both the DiPerna-Lions theory and adaptations of the
superposition principle, by many authors, including Le Bris and Lions [48],
Figalli [38], Trevisan [59], and Champagnat and Jabin [29]; see also the book
[49]. Just as in the first-order setting, the fact that the measure
$\operatorname{div}b$ may contain a singular part prevents the application of
these theories to the present situation.

In the compressive regime, we extend the viscosity solution theory of Section
3 to the second order equation

(1.9) $-\partial_{t}u+b(t,x)\cdot\nabla
u-\operatorname{tr}[a(t,x)\nabla^{2}u]=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad u(T,\cdot)=u_{T},$

where $b$ satisfies the one-sided Lipschitz condition (1.4) and $a$ is a
regular, but possibly degenerate, symmetric matrix. This equation, as well as
the dual problem

(1.10)
$\partial_{t}f-\operatorname{div}(b(t,x)f)-\nabla^{2}\cdot(a(t,x)f)=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad f(0,\cdot)=f_{0},$

can be related to the SDE

(1.11)
$d_{t}\Phi_{t,s}(x)=-b(t,\Phi_{t,s}(x))dt+\sigma(t,\Phi_{t,s}(x))dW_{t},\quad
t>s,\quad\Phi_{s,s}(x)=x,$

which is the SDE analogue of the backward flow for (1.3). Here $W$ is a given
Brownian motion and $a=\frac{1}{2}\sigma\sigma^{T}$. We establish the
existence and uniqueness, for every $x\in{\mathbb{R}}^{d}$, of a strong
solution in the Filippov sense, and we show that, with probability one,
$\Phi_{t,s}$ is Hölder continuous for any exponent less than $1$.

The situation is more complicated in the expansive regime, namely, for the
equations

(1.12) $-\partial_{t}u-b(t,x)\cdot\nabla
u-\operatorname{tr}[a(t,x)\nabla^{2}u]=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad u(T,\cdot)=u_{T}$

and

(1.13)
$\partial_{t}f+\operatorname{div}(b(t,x)f)-\nabla^{2}\cdot(a(t,x)f)=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad f(0,\cdot)=f_{0}.$

In the first-order setting, the characterization of the “good” distributional
solution of the continuity equation (1.8) relies on the Lipschitz continuity
of the backward ODE flow. Adapting similar methods for the second order
equation (1.13) involves establishing Lipschitz continuity of a stochastic
flow like (1.11) with certain time-reversed coefficients (see (4.30) below).
While flows of the form (1.11) are Hölder continuous for any exponent less
than $1$, it is an open question as to whether it is Lipschitz with
probability one. We relegate a general study of (1.12) and (1.13), and of the
stochastic regular Lagrangian flow for

(1.14)
$d_{t}\Phi_{t,s}(x)=b(t,\Phi_{t,s}(x))dt+\sigma(t,\Phi_{t,s}(x))dW_{t},\quad
t>s,\quad\Phi_{s,s}(x)=x,$

to future work. The exception333Another case of interest is when the diffusion
matrix $a$ is nondegenerate in which case very general results can be obtained
even for locally bounded $b$; see [38]. is when $\sigma$ is constant in the
${\mathbb{R}}^{d}$-variable. In this case, we prove that a suitable stochastic
flow of the form (1.11) can be inverted, leading, as in the deterministic
case, to the existence and uniqueness of a strong solution to (1.14) for a.e.
$x\in{\mathbb{R}}^{d}$, and a corresponding solution theory for the PDEs
(1.12) and (1.13).

###  1.2 Applications and further study

While interesting in their own right, linear transport equations and ODEs with
nonregular velocity fields arise naturally in several equations in fluid
dynamics, in which the velocity fields depend nonlinearly on various other
physical quantities that are coupled with the transported quantity. Since
these equations must be posed a priori in a weak sense, this leads to velocity
fields with limited regularity. The DiPerna-Lions and Ambrosio theories have
been successfully applied to a number of such situations; see [45, 13, 7, 34,
8, 9, 50, 62]. The one-dimensional Bouchut-James theory of reversible
solutions for transport equations with semi-Lipschitz velocity fields has been
successfully applied in applications to conservation laws and pressureless
gasses; see [22, 21, 40, 41].

Nonlinear transport equations also arise in certain models for large
population dynamics, specifically mean field games (MFG). In [46], the first
author and Lasry introduced a forward-backward system of PDEs modeling a large
population of agents in a state of Nash equilibrium. The evolution of the
density $f$ of players is described by a continuity equation (1.8) (or Fokker-
Planck equation (1.13)), where the velocity field $b$ is given by

(1.15) $b(t,x)=-\nabla_{p}H(t,x,\nabla u(t,x)).$

Here, $H$ is a convex Hamiltonian, and $u$ is the solution of the terminal
value problem

(1.16) $-\partial_{t}u-\operatorname{tr}[a(t,x)\nabla^{2}u]+H(t,x,\nabla
u(t,x))=F[f(t,\cdot)]\quad\text{in }(0,T)\times{\mathbb{R}}^{d},\quad
u(T,\cdot)=G[f(t,\cdot)],$

which is a Hamilton-Jacobi-Bellman equation encoding the optimization problem
for a typical agent, and whose influence by the population of agents is
described by the coupling functions $F$ and $G$. The velocity field (1.15) is
the consensus optimal feedback policy of the population of agents at a Nash
equilibrium.

When $a$ is degenerate, or even zero, the function $u$ has limited regularity,
and is no better than semiconcave in the spatial variable in general.
Therefore, even if $H$ is smooth, the velocity field (1.15) may satisfy at
most

(1.17) $b\in BV_{\mathrm{loc}}\quad\text{and}\quad(\operatorname{div}b)_{-}\in
L^{\infty}.$

This falls just outside the DiPerna-Lions-Ambrosio regime, since the measure
$(\operatorname{div}b)_{+}$ may still fail to be absolutely continuous in
general. In fact, the well-posedness of a suitable notion of solution for the
transport and ODE problems under the general assumptions (1.17) remains an
open problem.

Many simple but useful MFG models involve a linear-quadratic Hamiltonian of
the form

$H(t,x,p)=A(t,x)|p|^{2}+B(t,x)\cdot p+C(t,x)$

for smooth, real-valued $A,B,C$ with $A>0$. In this case, it is easy to see
that (1.15) satisfies the half-Lipschitz condition (1.4). This situation was
studied by Cardaliaguet and Souganidis [27] for first-order, stochastic mean
field games systems with common noise. In particular, it is proved there that
the uniqueness of probability density solutions of (1.7) gives rise, through
the superposition principle, to the uniqueness of optimal trajectories for the
probabilistic formulation of the MFG problem, and, moreover, the solution of
the stochastic forward-backward system can be used to construct approximate
Nash equilibria for the $N$-player game. Our analysis for the Fokker-Planck
equation (1.13) may therefore be expected to yield similar results for
stochastic MFG systems with common noise and degenerate, spatially-homogenous,
idiosyncratic noise, a special case of the equations considered by
Cardaliaguet, Souganidis, and the second author in [26].

The second application of nonlinear transport equations in mean field games is
involved with the master equation for a MFG with a finite state space. These
equations generally take the form

(1.18) $\partial_{t}u+b(t,x,u)\cdot\nabla u=c(t,x,u)\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},$

where $u$, $b$, and $c$ all take values in ${\mathbb{R}}^{d}$; coordinate-by-
coordinate, (1.18) is written as

$\partial_{t}u^{i}+b^{j}(t,x,u)\partial_{x_{j}}u^{i}=c^{i}(t,x,u),\quad
i=1,2,\ldots,d.$

Therefore, (1.18) is a nonconservative hyperbolic system, whose general well-
posedness is a difficult question in general; note that, when $d=1$, (1.18)
becomes a scalar conservation law.

We do not discuss (1.18) here, but, in a forthcoming paper [52], we study a
particular regime of equations taking the form (1.18), using a new theory for
linear transport equations with velocity fields $b$ that are increasing
_coordinate by coordinate_ , that is, $\partial_{x_{j}}b^{i}\geq 0$ for $i\neq
j$.

The extension to infinite dimensions, of both the linear problems (1.1)-(1.2),
as well as the nonlinear equation (1.18), remains an interesting question,
with numerous applications, including the study of mean field game master
equations on the Hilbert space of square-integrable random variables. We aim
to study these situations in future work.

###  1.3 Notation

Given a function space $X({\mathbb{R}}^{d})$, or $X(\Omega)$ for an
appropriate subdomain of ${\mathbb{R}}^{d}$, $X_{\mathrm{loc}}$ denotes the
space of functions (or distributions) $f$ such that $\phi f\in X$ for all
$\phi\in C^{\infty}_{c}({\mathbb{R}}^{d})$. If $X$ is a normed space, the same
is not necessarily true for $X_{\mathrm{loc}}$, but it inherits the topology
of local $X$-convergence. For example, $\lim_{n\to\infty}f_{n}=f$ in
$L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})$ means that
$\lim_{n\to\infty}\left\|f_{n}-f\right\|_{L^{p}(B_{R})}=0$ for all $R>0$. We
denote by $L^{p}_{+}([0,T])$ the subset of $L^{p}([0,T])$ consisting of
nonnegative functions.

Unless other specified, Banach or Fréchet spaces of functions are endowed with
the strong topology. For a function space $X$, the subscripts $X_{\mathrm{w}}$
and $X_{{\mathrm{w}\text{-}\star}}$ indicate the weak (resp. weak-$\star$)
topology.

For $1\leq p<\infty$, $\mathcal{P}_{p}$ is the space of probability measures
$\mu$, with $\int|x|^{p}\mu(dx)<\infty$, which becomes a complete metric space
for the $p$-Wasserstein distance $\mathcal{W}_{p}$.

The transpose of a matrix $\sigma$ is denoted by $\sigma^{T}$, and, if
$\sigma$ is a square matrix, its symmetric part is denoted by
$\mathrm{Sym}(\sigma):=\frac{1}{2}(\sigma+\sigma^{T})$. The symbol
$\operatorname{Id}$ stands for either the identity map or the identity matrix,
the precise meaning being clear from context.

##  2 The ODE flow

This section is focused on the solvability and properties of the flow
associated to a velocity field $b$ satisfying444The linear growth assumption
is a standard way to ensure that the a priori estimates for solutions do not
blow up. Otherwise, the results of the paper would need a corresponding local
theory, as for example in [10].

(2.1) $\left\\{\begin{split}&\text{for some }C_{0},C_{1}\in
L^{1}_{+}([0,T])\text{ and for all }t\in[0,T]\text{ and
}x,y\in{\mathbb{R}}^{d},\\\ &|b(t,x)|\leq C_{0}(t)(1+|x|)\quad\text{and}\\\
&(b(t,x)-b(t,y))\cdot(x-y)\geq-C_{1}(t)|x-y|^{2}.\end{split}\right.$

Because $b(t,\cdot)$ is not necessarily continuous, the ODE must be
interpreted in the Filippov sense [39], that is, abusing notation, we denote
by $b(t,x)$ the convex hull of all limit points of $b(t,y)$ as $y\to x$. For
$s\in[0,T]$, we seek absolutely continuous solutions $t\mapsto\phi_{t,s}(x)$
of the problem

(2.2) $\begin{dcases}\partial_{t}\phi_{t,s}(x)\in
b(t,\phi_{t,s}(x)),&t\in[0,T],\\\ \phi_{s,s}(x)=x.&\end{dcases}$

######  Remark 2.1.

If $\dot{X}(t)\in b(t,X(t))$ and

$\tilde{X}(t):=\exp\left(\int_{0}^{t}C_{1}(s)ds\right)X(t)\quad\text{and}\quad\tilde{b}(t,x):=\exp\left(\int_{0}^{t}C_{1}(s)ds\right)b\left(t,\exp\left(-\int_{0}^{t}C_{1}(s)dsx\right)\right),$

so that $\dot{\tilde{X}}\in\tilde{b}(t,\tilde{X}(t))$, then $\tilde{b}$
satisfies (2.1) with $C_{1}\equiv 0$ and a possibly different $C_{0}$. In
other words, with a change of variables, one may always assume $b$ is monotone
without loss of generality.

We sometimes use the following characterization and properties of half-
Lipschitz maps; see [23, Lemma 2.2].

######  Lemma 2.1.

A vector field $B:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfies

$(B(x)-B(y))\cdot(x-y)\geq-C|x-y|^{2}\quad\text{for some }C\geq 0\text{ and
all }x,y\in{\mathbb{R}}^{d}$

if and only if $\mathrm{Sym}(\nabla B)\geq-C\operatorname{Id}$ in the sense of
distributions. We then also have $B\in BD_{\mathrm{loc}}({\mathbb{R}}^{d})$,
and

$\mathrm{Sym}(\nabla B)-(\operatorname{tr}\nabla
B)\operatorname{Id}\leq(d-1)C\operatorname{Id}.$

The space $BD({\mathbb{R}}^{d})$ of bounded deformations is the space of
vector fields $B:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ such that the symmetric
part of the distribution $\nabla B$ is a locally bounded Radon measure, and,
as such, is a strictly larger space than $BV({\mathbb{R}}^{d})$. For more
details, see [58].

We fix a family of regularizations such that

(2.3) $\left\\{\begin{split}&(b^{\varepsilon})_{\varepsilon>0}\subset
L^{1}([0,T],C^{0,1}({\mathbb{R}}^{d})),\quad\lim_{\varepsilon\to
0}b^{\varepsilon}=b\text{ a.e. in }[0,T]\times{\mathbb{R}}^{d},\text{ and}\\\
&b^{\varepsilon}\text{ satisfies \eqref{A:monotoneB} uniformly in
$\varepsilon>0$}.\end{split}\right.$

For example, we may take
$b^{\varepsilon}(t,\cdot)=b(t,\cdot)*\rho_{\varepsilon}$ for
$\rho_{\varepsilon}=\varepsilon^{-d}\rho(\cdot/\varepsilon)$, with $\rho\in
C^{\infty}_{+}({\mathbb{R}}^{d})$, $\operatorname{supp}\rho\in B_{1}$, and
$\int\rho=1$.

###  2.1 The backward flow

We begin the analysis with the backward flow, that is, (2.2) for $t<s$. This
is the time-direction for which the one-sided Lipschitz condition (2.1) yields
a unique, Lipschitz flow. We record its properties here and refer to [39, 55,
56, 30] for the proofs; see also the work of Dafermos [35] for the connection
to generalized characteristics of conservation laws.

######  Lemma 2.2.

For every $(s,x)\in[0,T]\times{\mathbb{R}}^{d}$, there exists a unique
solution $\phi_{t,s}(x)$ of (2.2) defined for
$(t,x)\in[0,s]\times{\mathbb{R}}^{d}$, satisfying the Lipschitz bound

(2.4)
$|\phi_{t,s}(x)-\phi_{t,s}(y)|\leq\exp\left(\int_{t}^{s}C_{1}(r)dr\right)|x-y|\quad\text{for
all }0\leq t\leq s\leq T\text{ and }x,y\in{\mathbb{R}}^{d}.$

Moreover, there exists a constant $C>0$ depending only on $T$ and $C_{0}$ from
(2.1) such that

(2.5) $|\phi_{t,s}(x)|\leq C(|x|+1)\quad\text{for all }0\leq t\leq s\leq
T\text{ and }x\in{\mathbb{R}}^{d},$

and

(2.6) $\left\\{\begin{split}&|\phi_{t_{1},s}(x)-\phi_{t_{2},s,x}|\leq
C(1+|x|)|t_{1}-t_{2}|\quad\text{and}\\\
&|\phi_{t,s_{1}}(x)-\phi_{t,s_{2}}(x)|\leq C(1+|x|)|s_{1}-s_{2}|\\\ &\text{for
all }t_{1},t_{2}\in[0,s],\;s_{2},s_{2}\in[t,T],\text{ and
}x\in{\mathbb{R}}^{d}.\end{split}\right.$

For all $0\leq r\leq s\leq t\leq T$, $\phi_{r,s}\circ\phi_{s,t}=\phi_{r,t}$.
If $(b^{\varepsilon})_{\varepsilon>0}$ are regularizations satisfying (2.3),
then the corresponding backward flows $\phi^{\varepsilon}$ converge locally
uniformly as $\varepsilon\to 0$ to $\phi$.

######  Remark 2.2.

The a priori local boundedness and time-regularity estimates (2.5) and (2.6),
depending only on $C_{0}$ and not $C_{1}$, do not require the half-Lipschitz
assumption on $b(t,\cdot)$, and are therefore satisfied for any limiting
solutions of the ODE when $b$ satisfies the first condition in (2.1). On the
other hand, the half-Lipschitz assumption is crucial for the Lipschitz
continuity of the flow (2.4), as well as the uniqueness of the solution.

######  Remark 2.3.

Consider the backward flow in ${\mathbb{R}}$ corresponding to
$b(t,x)=b(x)=\operatorname{sgn}x$, which is given, for $x\in{\mathbb{R}}$ and
$s<t$, by

(2.7) $\phi_{s,t}(x)=\begin{dcases}x+(t-s)&\text{if }x<-t-s,\\\ 0&\text{if
}|x|\leq t-s,\text{ and}\\\ x-(t-s)&\text{if }x>t-s.\end{dcases}$

This demonstrates that, in general, the trajectories of the backward flow may
concentrate on sets of measures $0$, in particular, where $b$ has jump
discontinuities.

We will often consider the examples $b(x)=\operatorname{sgn}x$ in subsequent
parts of the paper in order to illustrate certain general phenomena and to
present counterexamples. Note that, by Remark 2.1, one can consider similar
examples for arbitrary $C_{1}\in L_{+}^{1}([0,1])$.

###  2.2 The Jacobian for the backward flow

In view of the Lipschitz regularity (2.4), $\nabla_{x}\phi_{t,s}\in
L^{\infty}$ for $t\leq s$, and so we can define the Jacobian

(2.8) $J_{t,s}(x):=\det(\nabla_{x}\phi_{t,s}(x))\quad\text{for }0\leq t\leq
s\leq T\quad\text{ and a.e. }x\in{\mathbb{R}}^{d}.$

######  Lemma 2.3.

Let $J$ be defined as in (2.8). Then $J\geq 0$,

(2.9) $\left\\{\begin{split}&J_{\cdot,s}\in
L^{\infty}([0,s]\times{\mathbb{R}}^{d})\cap
C([0,s],L^{1}_{\mathrm{loc}}({\mathbb{R}}^{d}))\quad\forall
s\in[0,T]\quad\text{and}\\\ &J_{t,\cdot}\in
L^{\infty}([t,T]\times{\mathbb{R}}^{d})\cap
C([t,T],L^{1}_{\mathrm{loc}}({\mathbb{R}}^{d}))\quad\forall
t\in[0,T],\end{split}\right.$ (2.10)
$\left\|J_{t,s}\right\|_{L^{\infty}}\leq\exp\left(d\int_{t}^{s}C_{1}(r)dr\right)\quad\text{for
all }0\leq t\leq s\leq T,$

and, for all $R>0$, there exists a modulus of continuity $\omega_{R}$, which
depends on $b$ only through the constants $C_{0}$ and $C_{1}$ in (2.1), such
that

(2.11)
$\left\\{\begin{split}&\left\|J_{t_{1},s}-J_{t_{2},s}\right\|_{L^{1}(B_{R})}\leq\omega_{R}(|t_{1}-t_{2}|)\quad\text{for
all }t_{1},t_{2}\in[0,s]\quad\text{and}\\\
&\left\|J_{t,s_{1}}-J_{t,s_{2}}\right\|_{L^{1}(B_{R})}\leq\omega_{R}(|s_{1}-s_{2}|)\quad\text{for
all }s_{1},s_{2}\in[t,T].\end{split}\right.$

If $(b^{\varepsilon})_{\varepsilon>0}$ are as in (2.3),
$(\phi^{\varepsilon})_{\varepsilon>0}$ are the corresponding solutions of
(2.2), and, for $\varepsilon>0$,
$J^{\varepsilon}=\det(\nabla_{x}\phi^{\varepsilon})$, then

(2.12) $\lim_{\varepsilon\to
0}J^{\varepsilon}_{\cdot,s}=J_{\cdot,s}\quad\text{weak-$\star$ in
}L^{\infty}([0,s]\times{\mathbb{R}}^{d})\quad\text{and}\quad\lim_{\varepsilon\to
0}J^{\varepsilon}_{t,\cdot}=J_{t,\cdot}\quad\text{weak-$\star$ in
}L^{\infty}([t,T]\times{\mathbb{R}}^{d}).$

###### Proof.

It suffices to prove all statements about $J_{\cdot,s}$ on $[0,s]$. The
arguments are exactly the same for the other halves using the fact that
$s\mapsto\phi_{t,s}$ is the forward flow corresponding to the velocity $-b$.

The convergence (2.12) goes through by compensated compactness arguments for
determinants; see the Appendix of [23]. The nonnegativity of $J$ now follows,
because $J^{\varepsilon}\geq 0$ for all $\varepsilon$.

For fixed $\varepsilon>0$ and $(s,x)\in[0,T]\times{\mathbb{R}}^{d}$, we have

$\partial_{t}J^{\varepsilon}_{t,s}(x)=\operatorname{div}_{x}b^{\varepsilon}(t,\phi^{\varepsilon}_{t,s}(x))J^{\varepsilon}_{t,s}(x)\quad\text{for
}t\in[0,s].$

Then (2.3) implies $\partial_{t}J^{\varepsilon}_{t,s}(x)\geq-
dC_{1}(t)J^{\varepsilon}_{t,s}(x)$, and so

$\frac{\partial}{\partial
t}\left(J^{\varepsilon}_{t,s}(x)e^{-d\int_{t}^{s}C_{1}(r)dr}\right)\geq 0.$

In particular, for $t_{1}<t_{2}\leq s$ and $R>0$,

$\int_{B_{R}}|J^{\varepsilon}_{t_{2},s}-J^{\varepsilon}_{t_{1},s}|\leq
e^{\int_{t_{1}}^{t_{2}}C_{1}(r)dr}\int_{B_{R}}J^{\varepsilon}_{t_{2},s}-\int_{B_{R}}J^{\varepsilon}_{t_{1},s}+\left(e^{\int_{t_{1}}^{t_{2}}C_{1}(r)dr}-1\right)\int_{B_{R}}J^{\varepsilon}_{t_{2},s}.$

Identifying the modulus of continuity $\omega_{R}$ in the statement of the
Lemma then reduces to proving the uniform-in-$\varepsilon$ continuity of

$[0,s]\ni t\mapsto\int_{B_{R}}J^{\varepsilon}_{t,s}(x)dx;$

note that $\int_{B_{R}}J^{\varepsilon}_{s,s}(x)dx=|B_{R}|$, so this will also
imply that $\int_{B_{R}}J^{\varepsilon}_{t,s}(x)dx$ is bounded uniformly in
$\varepsilon$.

In view of the $L^{\infty}$-boundedness of $J^{\varepsilon}$, it suffices to
prove the uniform-in-$\varepsilon$ continuity in $t$ of $\int
f(x)J^{\varepsilon}_{t,s}(x)dx$ for any $f\in C_{c}({\mathbb{R}}^{d})$. The
change of variables formula gives

$\int f(x)J^{\varepsilon}_{t,s}(x)dx=\int f(\phi^{\varepsilon}_{s,t}(x))dx.$

Note that
$\partial_{t}\phi^{\varepsilon}_{s,t}(x)=-b^{\varepsilon}(t,\phi^{\varepsilon}_{s,t}(x))$,
and the Lipschitz constant in $t$ of $\phi^{\varepsilon}_{s,t}(x)$ depends
only on an upper bound for $|x|$ and the constant $C_{0}$ in (2.1), and,
therefore, is independent of $\varepsilon$. ∎

When $d=1$, the $L^{\infty}$-weak-$\star$ convergence of
$J^{\varepsilon}=\partial_{x}\phi^{\varepsilon}$ to $J$ can be strengthened
via an Aubin-Lions type compactness result.

######  Proposition 2.1.

Assume $d=1$, and let $J^{\varepsilon}$ and $J$ be as in Lemma 2.3. Then

$\lim_{\varepsilon\to
0}J^{\varepsilon}_{\cdot,s}=J_{\cdot,s}\quad\text{strongly in
}L^{1}_{\mathrm{loc}}([0,s]\times{\mathbb{R}})\quad\text{and}\quad\lim_{\varepsilon\to
0}J^{\varepsilon}_{t,\cdot}=J_{t,\cdot}\quad\text{strongly in
}L^{1}_{\mathrm{loc}}([t,T]\times{\mathbb{R}}).$

###### Proof.

Fix $t\in[0,T]$ and $R>0$. Lemma 2.2 implies that there exists $M$ independent
of $\varepsilon$ such that $|\phi^{\varepsilon}_{t,s}(x)|\leq M$ for all
$s\in[t,T]$ and $x\in[-R,R]$. Upon redefining $b$ outside of
$[0,T]\times[-2R,2R]$, we find that $\phi_{t,s}(x)$, and therefore
$J_{t,s}(x)$, is unchanged, and therefore, in order to prove the
$L^{1}$-convergence in $[t,T]\times[-R,R]$, we may assume without loss of
generality that $b$ is bounded uniformly. Applying the transformation
$\tilde{\phi}_{t,s}(x)=\phi_{t,s}(x)-\int_{t}^{s}C(r)dr$ for an appropriate
$C\in L^{1}_{+}([0,T])$ depending on $C_{0}$ from (2.1), we may also assume
$b\geq 1$.

For $(s,x)\in[t,T]\times{\mathbb{R}}$, set
$f^{\varepsilon}(s,x)=J^{\varepsilon}_{t,s}(x)$. Then $f^{\varepsilon}$ solves
the continuity equation

$\partial_{s}f^{\varepsilon}+\partial_{x}\left(b^{\varepsilon}(s,x)f^{\varepsilon}\right)=0\quad\text{in
}[t,T]\times{\mathbb{R}}\quad\text{and}\quad f^{\varepsilon}(t,\cdot)=1.$

For a standard mollifier $\rho\in C^{\infty}_{c}([-1,1])$, let
$\rho_{n}=n\rho(\cdot/n)$ and $f^{\varepsilon,n}=\rho_{n}*_{t}f^{\varepsilon}$
be the mollification of $f^{\varepsilon}$ only in the time variable. We then
have

$\partial_{s}f^{\varepsilon,n}+\partial_{x}\left[\rho_{n}*_{t}(b^{\varepsilon}f^{\varepsilon})\right]=0\quad\text{in
}\left[t+\frac{1}{n},T\right]\times{\mathbb{R}}$

and, for any $R>0$,

$\sup_{s\in[t+1/n,T]}\left\|\partial_{x}\left[\rho_{n}*_{t}(b^{\varepsilon}f^{\varepsilon})\right](s,\cdot)\right\|_{L^{1}([-R,R])}\leq\sup_{s\in[t+1/n,T]}\left\|\partial_{s}f^{\varepsilon,n}(s,\cdot)\right\|_{L^{1}([-R,R])}\leq
n\left\|\rho^{\prime}\right\|_{L^{1}({\mathbb{R}})}\omega_{R}\left(\frac{1}{n}\right),$

where $\omega_{R}$ is as in (2.11). It follows that, for fixed
$n\in{\mathbb{N}}$,
$(\rho_{n}*_{t}(b^{\varepsilon}f^{\varepsilon}))_{\varepsilon>0}$ is
precompact in $L^{1}([t,T]\times[-R,R])$, and so, because

$\lim_{n\to\infty}\rho_{n}*_{t}(b^{\varepsilon}f^{\varepsilon})=bf$

in $L^{1}([t,T]\times[-R,R])$, uniformly in $\varepsilon$, we conclude that
$(b^{\varepsilon}f^{\varepsilon})_{\varepsilon>0}$ is precompact in
$L^{1}([t,T]\times[-R,R])$. This implies that, as $\varepsilon\to 0$,
$b^{\varepsilon}f^{\varepsilon}$ converges strongly in
$L^{1}([t,T]\times[-R,R])$ to $bf$.

Fix any subsequence $(\varepsilon_{n})_{n\geq 0}$ approaching zero as
$n\to\infty$. Then there exists a further subsequence such that
$f^{\varepsilon_{n_{k}}}b^{\varepsilon_{n_{k}}}\xrightarrow{k\to\infty}fb$
almost everywhere, and therefore
$f^{\varepsilon_{n_{k}}}\xrightarrow{k\to\infty}f$ a.e. in $[t,T]\times[-R,R]$
because $b\geq 1$ and

$\displaystyle f^{\varepsilon}(s,x)-f(s,x)$
$\displaystyle=\frac{b(s,x)f^{\varepsilon}(s,x)-b(s,x)f(s,x)}{b(s,x)}$
$\displaystyle=\frac{b^{\varepsilon}(s,x)f^{\varepsilon}(s,x)-b(s,x)f(s,x)}{b(s,x)}+\frac{\left(b(s,x)-b^{\varepsilon}(s,x)\right)f^{\varepsilon}(s,x)}{b(s,x)}.$

The convergence of $f^{\varepsilon_{n_{k}}}$ to $f$ in
$L^{1}([t,T]\times[-R,R])$, and therefore the convergence of the full family
$(f^{\varepsilon})_{\varepsilon>0}$ to $f$, is a consequence of the Lebesgue
dominated convergence theorem. ∎

######  Remark 2.4.

The one-dimensional structure is important in the proof of Proposition 2.1, in
particular, in deducing from the equicontinuity of $J^{\varepsilon}$ in time
that $(b^{\varepsilon}J^{\varepsilon})_{\varepsilon>0}$ belongs to a
precompact subset of $L^{1}$. It is not immediately clear whether this
argument can be extended to multiple dimensions.

###  2.3 The forward flow as the right-inverse of the backward flow

We next investigate the solvability of (2.2) forward in time. This is done by
analyzing the Jacobian $J$ from the previous subsection in order to invert the
backward flow. Similar methods are used in [27], and, by including the
Jacobian in the analysis, we obtain additionally the almost-everywhere
continuity of the inverse.

We will revisit this topic in Section 4 when we analyze the forward flow,
which will arise from the theory of renormalized solutions of the appropriate
transport equation.

######  Proposition 2.2.

For $t\leq s$, there exists a set $A_{ts}\subset{\mathbb{R}}^{d}$ of full
measure such that, for all $y\in A_{ts}$, $\phi_{t,s}^{-1}(\\{y\\})$ is a
singleton, which we denote by $\left\\{\phi_{s,t}(y)\right\\}$. Moreover,
there exists a version of the map
$\phi_{s,t}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ such that $\phi_{s,t}$ is
continuous a.e.

As an intermediate step, we first prove the following.

######  Lemma 2.4.

Assume $0\leq t\leq s\leq T$ and $K\subset{\mathbb{R}}^{d}$ is nonempty,
compact, and connected. Then $\phi_{t,s}^{-1}(K)$ is nonempty, compact, and
connected.

###### Proof.

For $r>0$, denote $K_{r}:=\bigcup_{y\in K}B_{r}(y)$. Fix a sequence
$(b^{n})_{n\in{\mathbb{N}}}$ satisfying (2.3)555That is, we abuse notation and
suppose that $b^{n}=b^{\varepsilon_{n}}$ for
$(b^{\varepsilon})_{\varepsilon>0}$ satisfying (2.3) and some
$(\varepsilon_{n})_{n\in{\mathbb{N}}}$ satisfying
$\lim_{n\to\infty}\varepsilon_{n}=0$., and let $\phi^{n}_{t,s}$ denote the
corresponding backward flow from the previous subsections.

We first show that

(2.13)
$\phi_{t,s}^{-1}(K)=\bigcap_{r>0}\bigcup_{n\in{\mathbb{N}}}\bigcap_{k\geq
n}(\phi_{t,s}^{k})^{-1}(K_{r}).$

Suppose $x\in\phi_{t,s}^{-1}(K)$. Then $y=\phi_{t,s}(x)\in K$. Setting
$y_{n}:=\phi^{n}_{t,s}(x)$, we have $\lim_{n\to\infty}y_{n}=y$ by Lemma 2.2,
which means that, for all $r>0$, there exists $n\in{\mathbb{N}}$ such that,
for all $k\geq n$, $\phi^{k}_{t,s}(x)\in B_{r}(y)\subset K_{r}$. This proves
the $\subset$ direction of (2.13).

Now suppose $x$ belongs to the right-hand side of (2.13). Then, for all $r>0$,
there exists $n\in{\mathbb{N}}$ such that $x\in(\phi_{t,s}^{k})^{-1}(K_{r})$
for all $k\geq n$. Set $y_{k}:=\phi_{t,s}^{k}(x)$, so that we have
$y_{k}\subset K_{r}$ for all $k\geq n$. We have
$y:=\lim_{k\to\infty}y_{k}=\lim_{k\to\infty}\phi_{t,s}^{k}(x)=\phi_{t,s}(x)$
by Lemma 2.2. On the other hand, we also have $y\in\overline{K_{r}}$, and so

$\phi_{t,s}(x)\subset\bigcap_{r>0}\overline{K_{r}}=K.$

Thus, the $\supset$ direction of (2.13) is established.

The continuity of $\phi_{t,s}$ and the compactness of $K$ imply that
$\phi_{t,s}^{-1}(K)$ is closed. We note also that
$(\phi^{k}_{t,s})^{-1}=\phi^{k}_{s,t}$ satisfies (2.5) uniformly in $k$,
because the bound only depends on the constant $C_{0}$ in the linear growth
bound of (2.1), which is also satisfied by $-b^{k}$. This along with (2.13)
implies that $\phi_{t,s}^{-1}(K)$ is bounded, and thus compact.

We now show that $\phi_{t,s}$ is surjective. Using again the bound (2.5)
satisfied uniformly in $k$ for $\phi^{k}_{s,t}$, we set
$x_{n}:=(\phi^{n}_{t,s})^{-1}(y)=\phi^{n}_{s,t}(y)$ and note that
$(x_{n})_{n\in{\mathbb{N}}}$ is bounded. Passing to a subsequence, we have
$\lim_{k\to\infty}x_{n_{k}}=x$ for some $x\in{\mathbb{R}}^{d}$, and then
$y=\phi^{n_{k}}_{t,s}(x_{k})$, so that
$y=\lim_{k\to\infty}\phi^{n_{k}}_{t,s}(x_{k})=\phi_{t,s}(x)$.

Finally, we show $\phi_{t,s}^{-1}(K)$ is connected. For each
$k\in{\mathbb{N}}$, $(\phi^{k}_{t,s})^{-1}(K_{r})$ is connected, and therefore
so is the intersection $\bigcap_{k\geq n}(\phi^{k}_{t,s})^{-1}(K_{r})$ for
each $n$. These sets are nested in $n$, so taking the union in
$n\in{\mathbb{N}}$ yields a connected set. Taking the intersection over $r>0$
gives the connectedness of $\phi^{-1}_{t,s}(K$). ∎

######  Remark 2.5.

The fact that the approximate backward flows converge _uniformly_ to
$\phi_{t,s}$ is used in the second-to-last paragraph of the proof, in order to
show that $\phi_{t,s}$ is surjective.

###### Proof of Proposition 2.2.

We identify the set by

$A_{t,s}=\left\\{y\in{\mathbb{R}}^{d}:\text{there exists
}x\in\phi_{t,s}^{-1}(\\{y\\})\text{ such that $\phi_{t,s}$ is differentiable
at $x$ and }J_{t,s}(x)\neq 0\right\\}.$

We first check that $A_{t,s}$ has full measure. Its complement consists of

$\displaystyle{\mathbb{R}}^{d}\backslash A_{ts}=$
$\displaystyle\left\\{y\in{\mathbb{R}}^{d}:J_{t,s}=0\text{ at the points of
differentiability of $\phi_{t,s}$ on }\phi_{t,s}^{-1}(\\{y\\})\right\\}$
$\displaystyle\cup\left\\{y\in{\mathbb{R}}^{d}:\phi_{t,s}\text{ is not
differentiable anywhere in }\phi_{t,s}^{-1}(\\{y\\})\right\\}.$

The fact that $\phi_{t,s}$ is differentiable a.e. and the change of variables
formula then give

$|{\mathbb{R}}^{d}\backslash
A_{t,s}|=\int_{{\mathbb{R}}^{d}}\mathbf{1}\\{\phi_{t,s}(x)\in{\mathbb{R}}^{d}\backslash
A_{t,s}\\}J_{t,s}(x)dx=0.$

It remains to show that $\phi_{t,s}^{-1}(\\{y\\})$ is a singleton for all
$y\in A_{t,s}$. By Lemma 2.4, $\phi_{t,s}^{-1}(\\{y\\})$ is nonempty, compact,
and connected. Suppose $x,\tilde{x}\in\phi_{t,s}^{-1}(\\{y\\})$ are such that
$J_{t,s}(x)\neq 0$. A Taylor expansion gives

$y=\phi_{t,s}(\tilde{x})=\phi_{t,s}(x)+\nabla_{x}\phi_{t,s}(x)\cdot(x-\tilde{x})+o(|x-\tilde{x}|)=y+\nabla_{x}\phi_{t,s}(x)\cdot(x-\tilde{x})+o(|x-\tilde{x}|).$

The invertibility of $\nabla_{x}\phi_{t,s}(x)$ then implies that, if
$|\tilde{x}-x|$ is sufficiently small, then $\tilde{x}=x$, or, in other words,
$x$ is an isolated point. But then the connected set $\phi_{t,s}(\\{y\\})$
must be equal to $\\{x\\}$, and we call $x=\phi_{s,t}(y)$.

For $y\in A_{t,s}$, we then have $(\phi_{t,s}\circ\phi_{s,t})(y)=y$. Since
$\phi_{t,s}^{-1}(\\{y\\})$ is nonempty for any $y\in{\mathbb{R}}^{d}$, we may
define a version of $\phi_{s,t}$ on all of ${\mathbb{R}}^{d}$ by imposing that
$\phi_{s,t}(y)\in(\phi_{t,s}^{-1})(\\{y\\})$ for any $y\in A_{t,s}$. For this
version, we have $\phi_{t,s}\circ\phi_{s,t}=\operatorname{Id}$ everywhere on
${\mathbb{R}}^{d}$. Suppose now that $y\in A_{t,s}$ and
$\lim_{n\to\infty}y_{n}=y$ for some sequence
$(y_{n})_{n\in{\mathbb{N}}}\subset{\mathbb{R}}^{d}$. Then

$\lim_{n\to\infty}(\phi_{t,s}\circ\phi_{s,t})(y_{n})=(\phi_{t,s}\circ\phi_{s,t})(y).$

We have

$(\phi_{s,t}(y_{n}))_{n\in{\mathbb{N}}}\subset\bigcup_{n\in{\mathbb{N}}}(\phi_{t,s})^{-1}(\\{y_{n}\\}),$

which implies by Lemma 2.4 that $(\phi_{s,t}(y_{n}))_{n\in{\mathbb{N}}}$ is
bounded. Letting $z$ be a limit point of this set, we have, by continuity of
the backward flow, that $y=\lim_{n\to\infty}y_{n}=\phi_{t,s}(z)$, and
therefore $z=\phi_{s,t}(y)$. ∎

######  Remark 2.6.

We shall see in Section 4 that the forward flow is always $BV$ in space.
Therefore, the “forward Jacobian” $J_{t,s}$ for $t>s$ can only be understood
as a measure. Indeed, returning to the example $b(t,x)=\operatorname{sgn}x$ on
${\mathbb{R}}$, the right inverse $\phi_{t,s}$ of $\phi_{s,t}$ given by (2.7)
is $\phi_{t,s}(x)=x+(\operatorname{sgn}x)(t-s)$ for $s\leq t$, which is
discontinuous only at $0$. The backward Jacobian is given by
$J_{s,t}(x)=\mathbf{1}\left\\{|x|\geq t-s\right\\}$, and the forward one is
$J_{t,s}=1+2(t-s)\delta_{0}$.

######  Remark 2.7.

The formula $\phi_{s,t}\circ\phi_{t,s}=\operatorname{Id}$ makes sense a.e. if
$s<t$, because $\phi_{s,t}$ is Lipschitz and $\phi_{t,s}$ is measurable. On
the other hand, $\phi_{t,s}$ is not also a left-inverse, since the formula
$\phi_{t,s}\circ\phi_{s,t}$ does not make sense. In the above example,
$\phi_{s,t}(x)$ is equal to $0$, for $|x|\leq t-s$, and $0$ is a point of
discontinuity for $\phi_{t,s}$. In general, the concentration of $\phi_{s,t}$
on sets of measure $0$ forbids applying $\phi_{t,s}$ as a left-inverse.

###  2.4 Compressive stochastic flows

We now fix a matrix-valued map

(2.14) $\Sigma\in L^{2}([0,T],C^{0,1}({\mathbb{R}}^{d};{\mathbb{R}}^{d\times
m})),$

and assume that

(2.15) $W:\Omega\times[0,T]\to{\mathbb{R}}^{m}\quad\text{is a standard
Brownian motion on a given probability space
}(\Omega,\mathcal{F},\mathbb{P},\mathbb{E}).$

In order to extend the results in the preceding subsections, and, in
particular, to bypass the difficulties of the backward time direction, we
consider forward SDEs with drift satisfying the opposite of (2.1), that is,

(2.16) $B:[0,T]\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d},\quad-B\text{
satisfies }\eqref{A:monotoneB},$

and consider the flow

(2.17)
$\begin{dcases}d_{s}\Phi_{s,t}(x)=B(s,\Phi_{s,t}(x))ds+\Sigma(s,\Phi_{s,t}(x))dW_{s},&s\in[t,T],\\\
\Phi_{t,t}(x)=x.\end{dcases}$

Once again, (2.17) must be understood in the Filippov sense, which means, for
$s\in[t,T]$,

(2.18)
$\Phi_{s,t}(x)=x+\int_{t}^{s}\alpha_{r}dr+\int_{t}^{s}\Sigma(r,\Phi_{r,t}(x))dW_{r},\quad\alpha_{s}\in
B(s,\Phi_{s,t}(x)),$

and we remark that our assumptions will allow us to always consider
probabilistically strong solutions; that is, we solve (2.18) path by path for
almost every continuous $W$ with respect to the Wiener measure. Depending on
the context in later sections (in particular, the time direction of
solvability for the transport and continuity equations), we consider different
examples for $B$ and $\Sigma$ for which these assumptions are satisfied.

######  Lemma 2.5.

For every $(t,x)\in[0,T]\times{\mathbb{R}}^{d}$ and ${\mathbb{P}}$-almost
surely, there exists a unique strong solution $\Phi_{s,t}(x)$ of (2.17)
defined on $[t,T]\times{\mathbb{R}}^{d}$. Moreover, for all $p\in[2,\infty)$,
there exists a constant $C=C_{p}>0$ depending only on the assumptions (2.1)
and (2.14) such that

(2.19) $\mathbb{E}|\Phi_{s,t}(x)-\Phi_{s,t}(y)|^{p}\leq
C|x-y|^{p}\quad\text{for all }0\leq t\leq s\leq T\text{ and
}x,y\in{\mathbb{R}}^{d},$ (2.20) ${\mathbb{E}}|\Phi_{s,t}(x)|^{p}\leq
C(|x|^{p}+1)\quad\text{for all }-0\leq t\leq s\leq T\text{ and
}x\in{\mathbb{R}}^{d},$

and

(2.21)
$\left\\{\begin{split}&{\mathbb{E}}|\Phi_{s_{1},t}(x)-\Phi_{s_{2},t}(x)|^{p}\leq
C(1+|x|)|s_{1}-s_{2}|^{p/2}\\\ &\text{for all $t\in[0,T]$,
$s_{1},s_{2}\in[t,T]$, and }x\in{\mathbb{R}}^{d}.\end{split}\right.$

With probability one, for all $0\leq r\leq s\leq t\leq T$,
$\Phi_{t,s}\circ\Phi_{s,r}=\Phi_{t,r}$. If $(b^{\varepsilon})_{\varepsilon>0}$
are regularizations satisfying (2.3), then, with probability one, the
corresponding stochastic flows $\Phi^{\varepsilon}$ converge locally uniformly
as $\varepsilon\to 0$ to $\Phi$.

###### Proof.

For $\varepsilon>0$, let $B^{\varepsilon}$ be the convolution of $B$ in space
by a standard mollifier (so that $b^{\varepsilon}:=-B^{\varepsilon}$ satisfies
(2.3)), and let $\Phi^{\varepsilon}_{t,s}$ denote the corresponding stochastic
flow. Itô’s formula, the one-sided Lipschitz assumption, and the Lipschitz
continuity of $\Sigma$ yield, for any $p\geq 2$ and some $C\in
L^{1}_{+}([0,T])$,

$\displaystyle\frac{\partial}{\partial
t}{\mathbb{E}}|\Phi^{\varepsilon}_{t,s}(x)-\Phi^{\varepsilon}_{t,s}(y)|^{p}\leq
C(t){\mathbb{E}}|\Phi^{\varepsilon}_{t,s}(x)-\Phi^{\varepsilon}_{t,s}(y)|^{p},$

which, along with Grönwall’s inequality, leads to the first statement. The
other two estimates are proved similarly, with constants independent of
$\varepsilon>0$.

In view of (2.19) and (2.21), the Kolmogorov continuity criterion then yields,
for any $R>0$, $p\geq 2$ and $\delta\in(0,1)$, a constant $C=C_{R,p,\delta}>0$
such that, for all $s\in[0,T]$, $\lambda\geq 1$ and $\varepsilon>0$,

${\mathbb{P}}\left(\sup_{x,y\in
B_{R}}\sup_{r,s\in[s,T]}\frac{|\Phi^{\varepsilon}_{t,s}(x)-\Phi^{\varepsilon}_{r,s}(y)|}{|x-y|^{1-\delta}+|t-s|^{\frac{1}{2}(1-\delta)}}>\lambda\right)\leq\frac{C}{\lambda^{p}}.$

It follow that the probability measures on
$C([s,T]\times{\mathbb{R}}^{d};{\mathbb{R}}^{d})$ induced by the random
variables $(\Phi^{\varepsilon}_{\cdot,s})_{\varepsilon>0}$ are tight with
respect to the topology of locally uniform convergence, and therefore converge
weakly along a subsequence as $\varepsilon\to 0$ to a probability measure that
gives rise to a weak (in the probabilistic sense) solution of (2.17), for
which the estimates in the statement of the lemma continue to hold.

A similar computation to the one above reveals that, for a fixed probability
space and almost every Brownian path $W$, the solution of (2.17) is unique.
The pathwise uniqueness then implies, by a standard argument due to Yamada and
Watanabe [61], that there is a unique strong solution for every
$x\in{\mathbb{R}}^{d}$. ∎

######  Remark 2.8.

It is an open question whether $\Phi_{t,s}$ is Lipschitz continuous, even if
$B$ is Lipschitz. When $B$ is Lipschitz and $\Sigma\in C^{1,\alpha}$ for some
$\alpha\in(0,1]$, it turns out the flow $\Phi_{t,s}$ is
$C^{1,\alpha^{\prime}}$ for any $\alpha^{\prime}\in(0,\alpha)$, but it is not
clear how to extend this to the case where $-B$ satisfies the one-sided
Lipschitz bound from below.

As a consequence, an understanding of the Jacobian
$\det(\nabla_{x}\Phi_{t,s}(x))$, or of the stability with respect to
regularizations of $B$, is considerably more complicated in the stochastic
case. The results of Section 4, where we discuss the expansive regime, are
therefore constrained to the first-order case, and we relegate the second-
order analysis to future work. One exception is when $\Sigma$ is independent
of the spatial variable, in which case a change of variables relates the SDE
to an ODE of the form (2.2) with a random $b$.

###  2.5 Small noise approximations

We return to the backward flow $\phi_{t,s}$, $0\leq t\leq s\leq T$, from Lemma
2.2. Recall that the backward flow also corresponds to the forward flow for
$-b$; that is,

(2.22) $\frac{\partial}{\partial s}\phi_{t,s}(x)=-b(s,\phi_{t,s}(x)),\quad
s\geq t,\quad\phi_{t,t}(x)=x.$

For $\varepsilon>0$, let $\phi^{\varepsilon}_{t,s}(x)$ denote the following
stochastic flow

(2.23)
$d_{s}\phi^{\varepsilon}_{t,s}(x)=-b(s,\phi^{\varepsilon}_{t,s}(x))ds+\varepsilon
dW_{s}\quad s\geq t,\quad\phi^{\varepsilon}_{t,t}(x)=x,$

where $W$ is now a $d$-dimensional Brownian motion. We note that (2.17) falls
under the assumptions of Lemma 2.5, but in fact (2.23) admits a unique strong
solution as soon as $b$ is merely locally bounded [36, 60]. In general, the
limiting solutions as $\varepsilon\to 0$ are not unique; however, we
immediately have the following as a consequence of Lemma 2.5.

######  Proposition 2.3.

For every $\varepsilon>0$, there exists a unique strong solution of (2.23).
Moreover, as $\varepsilon\to 0$, $\phi^{\varepsilon}$ converges locally
uniformly to $\phi$.

If $J^{\varepsilon}=\det(\nabla_{x}\phi^{\varepsilon})$, then, as
$\varepsilon\to 0$, $J^{\varepsilon}$ converges weak-$\star$ in
$L^{\infty}([s,T]\times{\mathbb{R}}^{d})$ and weakly in
$C([s,T],L^{1}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ to $J$.

##  3 The compressive regime

In this section, we consider the transport and continuity equations in the so-
called compressive regime. That is, for velocity field $b$ satisfying (2.1),
we study the TVP for the nonconservative equation

(3.1) $-\frac{\partial u}{\partial t}+b(t,x)\cdot\nabla u=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}\quad\text{and}\quad u(T,\cdot)=u_{T}\quad\text{in
}{\mathbb{R}}^{d},$

and the IVP for the conservative equation

(3.2) $\frac{\partial f}{\partial
t}-\operatorname{div}(b(t,x)f)=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}\quad\text{and}\quad f(0,\cdot)=f_{0}.$

We recall that $\operatorname{div}b$ is bounded from below, and therefore, the
direction of time for (3.1) and (3.2) does not allow for a solution theory in
Lebesgue spaces, due to the concentrative nature of the backward flow analyzed
in the previous section. The TVP (3.1) will be solved in the space of
continuous functions, while (3.2) is solved in the dual space of locally
bounded Radon measures.

We also obtain analogous result for the second-order equations

(3.3) $-\frac{\partial u}{\partial
t}-\operatorname{tr}[a(t,x)\nabla^{2}u]+b(t,x)\cdot\nabla u=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}\quad\text{and}\quad u(T,\cdot)=u_{T}$

and

(3.4) $\frac{\partial f}{\partial
t}-\operatorname{div}\big{[}\operatorname{div}(a(t,x)f)-b(t,x)f\big{]}=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}\quad\text{and}\quad f(0,\cdot)=f_{0},$

where $a=\frac{1}{2}\sigma\sigma^{T}$ for
$\sigma:[0,T]\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times m}$ satisfying

(3.5)
$\sup_{x\in{\mathbb{R}}^{d}}\frac{|\sigma(\cdot,x)|}{1+|x|}+\sup_{y,z\in{\mathbb{R}}^{d}}\frac{|\sigma(\cdot,y)-\sigma(\cdot,z)|}{|y-z|}\in
L^{2}([0,T]).$

###  3.1 The nonconservative equation

####  3.1.1 Representation formula

When interpreting (3.1) in the distributional sense, we are constrained to
seek solutions that are continuous. Indeed, the distribution

$b\cdot\nabla u=\operatorname{div}(bu)-(\operatorname{div}b)u$

pairs the solution $u$ with $\operatorname{div}b$, which is a measure in
general. The other motivating factor is the formal representation formula for
the solution of the TVP (3.1), which is given in terms of the backward flow:

(3.6) $u(t,x)=u_{T}(\phi_{t,T}(x))\quad\text{for
}(t,x)\in[0,T]\times{\mathbb{R}}^{d}.$

This formula and the Lipschitz continuity of $\phi_{t,T}$ given in Lemma 2.2
suggest that the solution operator for (3.1) should preserve continuity. In
fact, the formula (3.6) defines a distributional solution, which is uniquely
obtained from limits of natural regularizations of the equation.

######  Theorem 3.1.

If $u_{T}\in C({\mathbb{R}}^{d})$, then the function $u$ in (3.6) is a
distributional solution of (3.1). Moreover, if
$(b^{\varepsilon})_{\varepsilon>0}$ satisfy (2.3) and $u^{\varepsilon}$ is the
corresponding solution of (3.1) with velocity field $b^{\varepsilon}$, then,
as $\varepsilon\to 0$, $u^{\varepsilon}$ converges locally uniformly to $u$.

###### Proof.

The unique solution $u^{\varepsilon}$ for the regularized velocity field is
given by $u^{\varepsilon}(t,\cdot)=u_{T}\circ\phi^{\varepsilon}_{t,T}$, where
$\phi^{\varepsilon}$ is the flow corresponding to $b^{\varepsilon}$. By Lemma
2.2, as $\varepsilon\to 0$, $\phi^{\varepsilon}$ converges locally uniformly
to $\phi$, and so the local-uniform convergence to $u$ follows from the
continuity of $u_{0}$.

Multiplying the equation for $u^{\varepsilon}$ by some $\psi\in
C^{1}_{c}((0,T)\times{\mathbb{R}}^{d})$ and integrating by parts gives

$\int_{0}^{T}\int_{{\mathbb{R}}^{d}}u^{\varepsilon}(t,x)\left(\partial_{t}\psi(t,x)-b^{\varepsilon}(t,x)\cdot\nabla\psi(t,x)+(\operatorname{div}b^{\varepsilon}(t,x))\psi(t,x)\right)dxdt=0.$

As $\varepsilon\to 0$, $b^{\varepsilon}\to b$ almost everywhere and
$\operatorname{div}b^{\varepsilon}\rightharpoonup\operatorname{div}b$ weakly
in the sense of measures, and so the fact that $u$ is a distributional
solution follows. ∎

Turning next to the second-order equation (3.3), we identify a solution
candidate with the appropriate stochastic flow. We do so by changing the time
direction in $b$ and $\sigma$ and considering the SDE

(3.7)
$\begin{dcases}d_{s}\Phi_{s,t}(x)=-b(s,\Phi_{s,t}(x))ds+\sigma(s,\Phi_{s,t}(x))dW_{s},&s\in[t,T],\\\
\Phi_{t,t}=\operatorname{Id},\end{dcases}$

where $W$ is as in (2.15). Note that (3.7) is of the type in (2.17) and thus
falls within the assumptions of Lemma 2.5. In particular, if $u_{T}$ is
continuous, then, in view of (2.19)-(2.21), the formula

(3.8) $u(t,x)={\mathbb{E}}[u_{T}(\Phi_{T,t}(x))]$

defines a continuous function. Moreover, if $u_{T}$ is Lipschitz, then
$u(t,\cdot)$ is Lipschitz for all $t>0$, and $1/2$-Hölder continuous in time.
Note that, in this case, the distribution
$\operatorname{tr}[a\nabla^{2}u]=\operatorname{div}(a\nabla
u)-\operatorname{div}a\cdot\nabla u$ makes sense, because $\nabla u$ and
$\operatorname{div}a$ both belong to $L^{\infty}$.

The following is proved exactly as for Theorem 3.1, with the use of the
estimates in Lemma 2.5.

######  Theorem 3.2.

Let $u_{T}\in C({\mathbb{R}}^{d})$ and define $u$ by (3.8). If
$(b^{\varepsilon})_{\varepsilon>0}$ satisfy (2.3) and $u^{\varepsilon}$ is the
corresponding solution of (3.3) with velocity $b^{\varepsilon}$, then, as
$\varepsilon\to 0$, $u^{\varepsilon}$ converges locally uniformly to $u$.
Moreover, if $u_{T}\in C^{0,1}$, then

$\sup_{(t,x,y)\in[0,T]\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{|u(t,x)-u(t,y)|}{|x-y|}+\sup_{(r,s,z)\in[0,T]\times{\mathbb{R}}^{d}}\frac{|u(r,z)-u(s,z)|}{|r-s|^{1/2}(1+|z|)}<\infty,$

and $u$ is a distributional solution of (3.3).

As a special case, we consider, for $\varepsilon>0$, the “viscous” version of
(3.1), that is

(3.9) $-\partial_{t}u^{\varepsilon}-\frac{\varepsilon^{2}}{2}\Delta
u^{\varepsilon}+b(t,x)\cdot\nabla u^{\varepsilon}=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad u^{\varepsilon}(T,\cdot)=u_{T}.$

This uniformly parabolic equation has a unique classical solution for any
$u_{T}\in C({\mathbb{R}}^{d})$, which, moreover, is given by
$u^{\varepsilon}(t,x)={\mathbb{E}}[u_{T}(\phi^{\varepsilon}_{t,T}(x))]$, where
now $\phi^{\varepsilon}$ denotes the solution of the SDE (2.23) from the
previous section. Arguing just as in Theorem 3.1 and invoking Proposition 2.3
immediately gives the following.

######  Theorem 3.3.

As $\varepsilon\to 0$, the solution $u^{\varepsilon}$ converges locally
uniformly to the function $u$ given by (3.6).

####  3.1.2 Viscosity solutions

Although (3.6) and (3.8) are the distributional solutions that arise uniquely
through regularization (either of $b$ or through vanishing viscosity limits),
it turns out that distributional solutions are not unique in general (see
subsubsection 3.1.3 below). It is then a natural question as to whether the
“good” solutions can be characterized other than as limits of regularizations,
or by the explicit formulae. For example, this is done for the one-dimensional
problem in [55] by introducing a sort of entropy condition.

We give a different characterization here using the theory of viscosity
solutions [32], which covers both the first- and second-order problems. We
present the results here only in the second-order case, which includes the
first-order equations when $a=0$.

We define, for $(t,x,p)\in[0,T]\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$,

$\underline{b}(t,x,p)=\liminf_{z\to x}b(t,z)\cdot
p\quad\text{and}\quad\overline{b}(t,x,p)=\limsup_{z\to y}b(t,z)\cdot p.$

For fixed $(t,x)\in[0,T]\times{\mathbb{R}}^{d}$, $\underline{b}(t,x,\cdot)$
and $\overline{b}(t,x,\cdot)$ are Lipschitz continuous on ${\mathbb{R}}^{d}$,
and, for fixed $(t,p)\in[0,T]$, $\underline{b}(t,\cdot,p)$ and
$\overline{b}(t,\cdot,p)$ are respectively lower and upper semicontinuous.

The following definition of viscosity (sup, super) solutions closely resembles
the one in [51].

######  Definition 3.1.

An upper-semicontinuous (resp. lower-semicontinuous) function $u$ is called a
subsolution (resp. supersolution) of (3.3) if, for all
$\psi:[0,T]\times{\mathbb{R}}^{d}$ that are $C^{1}$ in $t$ and $C^{2}$ in $x$,
it holds that

$-\frac{d}{dt}\max_{x\in{\mathbb{R}}^{d}}\left\\{u(t,x)-\psi(t,x)\right\\}\leq\inf\left\\{\operatorname{tr}[a(t,y)\nabla^{2}\psi(t,y)]-\underline{b}(t,y,\nabla\psi(t,y)):y\in\operatorname*{arg\,max}\\{u(t,\cdot)-\psi(t,\cdot)\\}\right\\}$

(resp.

$-\frac{d}{dt}\min_{x\in{\mathbb{R}}^{d}}\left\\{u(t,x)-\psi(t,x)\right\\}\geq\sup\left\\{\operatorname{tr}[a(t,y)\nabla^{2}\psi(t,y)]-\overline{b}(t,y,\nabla\psi(t,y)):y\in\operatorname*{arg\,min}\\{u(t,\cdot)-\psi(t,\cdot)\\}\right\\}\Big{)}.$

If $u\in C([0,T]\times{\mathbb{R}}^{d})$ is both a sub and supersolution, we
say $u$ is a solution.

The comparison principle is proved by doubling the space variable. In
particular, we have the following lemma, which follows exactly by methods as
in [31, 53, 54]. For
$(t,x,y)\in[0,T]\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$, we define the
nonnegative matrix

$A(t,x,y):=\begin{pmatrix}\sigma(t,x)\\\
\sigma(t,y)\end{pmatrix}\begin{pmatrix}\sigma(t,x)^{T}&\sigma(t,y)^{T}\end{pmatrix}.$

######  Lemma 3.1.

Assume $u$ and $v$ are respectively a sub and supersolution of (3.3). Then
$w(t,x,y)=u(t,x)-v(t,y)$ is a subsolution of

$-\partial_{t}w-\operatorname{tr}[a(t,x,y)\nabla^{2}_{(x,y)}w]+\underline{b}(t,x,\nabla_{x}w)-\overline{b}(t,x,-\nabla_{y}w)\leq
0.$

We may now state and prove the comparison principle.

######  Theorem 3.4.

If $u$ and $v$ are respectively a sub and supersolution of (3.3) such that

$\sup_{(t,x)\in[0,T]\times{\mathbb{R}}^{d}}\frac{u(t,x)}{1+|x|}+\sup_{(s,y)\in[0,T]\times{\mathbb{R}}^{d}}\frac{-v(s,y)}{1+|y|}<\infty,$

then $t\mapsto\sup_{x\in{\mathbb{R}}^{d}}\left\\{u(t,x)-v(t,x)\right\\}$ is
nondecreasing.

###### Proof.

Define $w(t,x,y):=u(t,x)-v(t,y)$, fix $\delta,\varepsilon>0$, and define
$\Phi_{\delta,\varepsilon}(x,y)=\frac{1}{2\delta}|x-y|^{2}+\frac{1}{2\varepsilon}(|x|^{2}+|y|^{2})$.
In view of the growth of $u$ and $v$ in $x$, for all $t\in[0,T]$, the map
$w(t,\cdot,\cdot)-\Phi_{\delta,\varepsilon}(x,y)$ attains a maximum on
${\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$. Moreover, standard arguments from
the theory of viscosity solutions (see for instance [32, Lemma 3.1]) imply
that there exist $\rho_{\delta}>0$ and $\lambda_{\varepsilon}$ such that
$\lim_{\delta\to 0}\rho_{\delta}^{2}/\delta=\lim_{\varepsilon\to
0}\varepsilon\lambda_{\varepsilon}^{2}=0$, and

$|x-y|\leq\rho_{\delta}\quad\text{and}\quad|x|+|y|\leq\lambda_{\varepsilon}\quad\text{for
all
}(x,y)\in\operatorname*{arg\,max}\left\\{w(t,\cdot,\cdot)-\Phi_{\delta,\varepsilon}\right\\},\quad
t\in[0,T].$

Therefore, if $t\in[0,T]$ and
$(x,y)\in\operatorname*{arg\,max}\left\\{w(t,\cdot,\cdot)-\Phi_{\delta,\varepsilon}\right\\}$,
we have, for some $C\in L^{1}_{+}([0,T])$,

$\displaystyle\operatorname{tr}$
$\displaystyle[a(t,x,y)\nabla^{2}_{(x,y)}\Phi_{\delta,\varepsilon}(x,y)]$
$\displaystyle=\operatorname{tr}\left[\left(\frac{1}{\delta}\begin{pmatrix}\operatorname{Id}&-\operatorname{Id}\\\
-\operatorname{Id}&\operatorname{Id}\end{pmatrix}+\varepsilon\begin{pmatrix}\operatorname{Id}&0\\\
0&\operatorname{Id}\end{pmatrix}\right)\begin{pmatrix}\sigma(t,x)\\\
\sigma(t,y)\end{pmatrix}\begin{pmatrix}\sigma(t,x)^{T}&\sigma(t,y)^{T}\end{pmatrix}\right]$
$\displaystyle\leq
C(t)\left(\frac{\rho_{\delta}^{2}}{\delta}+\varepsilon\lambda_{\varepsilon}^{2}\right)$

and

$\displaystyle-\underline{b}$
$\displaystyle\left(t,x,\nabla_{x}\Phi_{\delta,\varepsilon}(x,y)\right)+\overline{b}\left(t,y,-\nabla_{y}\Phi_{\delta,\varepsilon}(x,y)\right)$
$\displaystyle=\limsup_{(z,w)\to(x,y)}\left\\{-b(t,z)\cdot\left(\frac{x-y}{\delta}+\beta
x\right)+b(t,w)\cdot\left(\frac{x-y}{\delta}-\beta y\right)\right\\}$
$\displaystyle=\limsup_{(z,w)\to(x,y)}\left\\{-(b(t,z)-b(t,w))\cdot\frac{z-w}{\delta}-b(t,z)\cdot\beta
z+b(t,w)\cdot\beta w\right\\}$ $\displaystyle\leq
C(t)\left(\frac{\rho_{\delta}^{2}}{\delta}+\varepsilon+\varepsilon\lambda_{\varepsilon}\right).$

It now follows from Definition 3.1 and Lemma 3.1 that, for some
$C_{\delta,\varepsilon}\in L^{1}_{+}([0,T])$ satisfying
$\lim_{(\delta,\varepsilon)\to(0,0)}C_{\delta,\varepsilon}=0$ in
$L^{1}([0,T])$,

$t\mapsto\sup_{(x,y)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\left\\{w(t,x,y)-\Phi_{\delta,\varepsilon}(x,y)\right\\}-\int_{t}^{T}C_{\delta,\varepsilon}(s)ds$

is nondecreasing. The result follows upon sending $\delta$ and $\varepsilon$
to $0$. ∎

As a consequence of the comparison theorem, the “good” distributional solution
of (3.3) can be uniquely characterized.

######  Theorem 3.5.

Assume $u_{T}\in C({\mathbb{R}}^{d})$ and $u_{T}\cdot(1+|\cdot|^{-1})\in
L^{\infty}$. Then (3.8) is the unique viscosity solution of (3.3).

###### Proof.

The fact that (3.8) defines a viscosity solution is due to Theorem 3.2 and the
stability properties of viscosity solutions666Note that smooth solutions of
the equation corresponding to $b^{\varepsilon}$ satisfying (2.3), or of the
viscous equation (3.9), are viscosity solutions in the sense of Definition
3.1.. In view of Lemma 2.5 and the growth of $u_{T}$, we may appeal to Theorem
3.4 to conclude that (3.8) is the only viscosity solution of the terminal
value problem (3.3). ∎

####  3.1.3 (Non)equivalence of distributional and viscosity solutions

For $x\in{\mathbb{R}}$, set $b(t,x)=\operatorname{sgn}x$ and $u_{T}(x)=|x|$.
Using the formula (2.7) for the backward flow, the solution (3.6) becomes

(3.10) $u(t,x)=(|x|-(T-t))_{+}.$

However, the Lipschitz function

(3.11) $v(t,x)=|x|-(T-t)$

is another distributional solution (and in fact satisfies the equation a.e.).
It can also be checked directly that (3.11) does not give a viscosity solution
of (3.1). Indeed, note that $v(t,x)-t$ attains a global minimum at any
$(t,0)\in[0,T]\times{\mathbb{R}}$. Applying the supersolution definition with
$\phi(t,x)=t$ yields the contradictory $-1\geq 0$.

The uniqueness of distributional solutions fails even if $b$ is continuous.
Indeed, if $0<\alpha<1$ and $b(t,x)=\operatorname{sgn}x|x|^{\alpha}$ and
$u_{T}(x)=|x|^{1-\alpha}$, then, arguing similarly as in the above example,

(3.12) $u(t,x)=\left(|x|^{1-\alpha}-(1-\alpha)(T-t)\right)_{+}$

and

(3.13) $v(t,x)=|x|^{1-\alpha}-(1-\alpha)(T-t)$

are two distributional solutions, and (3.12) is the one corresponding to
(3.6). Once again, (3.13) can directly be seen to fail the viscosity
supersolution property.

In the first example above, $u_{T}$ is Lipschitz while $b$ is discontinuous,
and, while $b$ is continuous in the second example, we take $u_{T}$ to be non-
Lipschitz. This should be compared with the following sufficient criterion for
equivalence.

######  Theorem 3.6.

If $b\in C([0,T]\times{\mathbb{R}}^{d})$ satisfies (2.1) and $u_{T}\in
C^{0,1}({\mathbb{R}}^{d})$, then there exists a unique distributional solution
$u\in C([0,T],C^{0,1}({\mathbb{R}}^{d}))$ given by (3.6).

###### Proof.

Let $\rho\in C_{c}^{\infty}$ be a standard mollifier and, for $\varepsilon>0$,
set $\rho_{\varepsilon}(x)=\varepsilon^{-d}\rho(\varepsilon^{-1}x)$. Let $u\in
C([0,T],C^{0,1}({\mathbb{R}}^{d}))$ be a distributional solution of (3.1) and
define $u_{\varepsilon}=u*\rho_{\varepsilon}$. Then

(3.14) $-\partial_{t}u_{\varepsilon}+b\cdot\nabla
u_{\varepsilon}=r_{\varepsilon}\quad\text{in }(0,T)\times{\mathbb{R}}^{d},$

where

$r_{\varepsilon}(t,x)=\int_{{\mathbb{R}}^{d}}\left(b(t,y)-b(t,x)\right)\cdot\nabla
u(t,y)\rho_{\varepsilon}(x-y)dy.$

Note that $r_{\varepsilon}\in C([0,T]\times{\mathbb{R}}^{d})$, and
$u_{\varepsilon}$ solves (3.14) in the sense of viscosity solutions. Moreover,
the continuity of $b$ and boundedness of $\nabla u$ imply that
$r_{\varepsilon}\xrightarrow{\varepsilon\to 0}0$ locally uniformly. Standard
stability results from the theory of viscosity solutions then imply that the
limit $u$ of $u_{\varepsilon}$ is the unique viscosity solution of (3.1). ∎

The above result can be extended by studying the interplay between regularity
of $b$ and $u$.

######  Theorem 3.7.

Suppose that $\alpha,\beta\in(0,1]$ satisfy $\alpha+\beta>1$, $b$ satisfies
(2.1) and $\sup_{t\in[0,T]}[b(t,\cdot)]_{C^{\alpha}}<\infty$, and $u$ is a
distributional solution of (3.1) such that
$\sup_{t\in[0,T]}[u(t,\cdot)]_{C^{\beta}}<\infty$. Then $u$ is the unique
viscosity solution of (3.1).

######  Remark 3.1.

The condition on $\alpha+\beta$, and, in particular, the strict inequality, is
sharp, as the example above with $b(x)=\operatorname{sgn}x|x|^{\alpha}$ and
$u_{T}(x)=|x|^{1-\alpha}$ shows.

###### Proof of Theorem 3.7.

Arguing similarly as for Theorem 3.6, it suffices to prove that

$r_{\varepsilon}=(b\cdot\nabla
u)*\rho_{\varepsilon}-b\cdot\nabla(u*\rho_{\varepsilon})\xrightarrow{\varepsilon\to
0}0\quad\text{locally uniformly},$

where $\rho_{\varepsilon}$ is a standard mollifier. We note that
$r_{\varepsilon}=M_{\varepsilon}[b(t,\cdot),u(t,\cdot)]$, where the bilinear
operator $M_{\varepsilon}$ is defined, for sufficiently regular
$(B,U):{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}\times{\mathbb{R}}$, by

$M_{\varepsilon}[B,U]=\int_{{\mathbb{R}}^{d}}\left(B(y)-B(x)\right)\cdot\nabla
U(y)\rho_{\varepsilon}(x-y)dy.$

Standard interpolation arguments give, for some $C>0$ depending on $\alpha$
and $\beta$, for all $(B,U)\in C^{\alpha}\times C^{\beta}$,

$\left|M_{\varepsilon}[B,U]\right|\leq
C\varepsilon^{\alpha+\beta-1}[B]_{C^{\alpha}}[U]_{C^{\beta}}.$

Therefore $|r_{\varepsilon}(t,x)|\leq
C[b(t,\cdot)]_{C^{\alpha}}[u(t,\cdot)]_{C^{\beta}}\varepsilon^{\alpha+\beta-1}$,
and we conclude upon sending $\varepsilon\to 0$. ∎

###  3.2 The conservative equation

####  3.2.1 Duality solutions

For either of the two conservative equations (3.2) and (3.4), the tendency of
the backward flow to concentrate on sets of Lebesgue measure zero implies
that, even if $f_{0}$ is absolutely continuous with respect to the Lebesgue
measure, $f(t,\cdot)$ may develop a singular part for $t>0$.

This presents an obstacle in defining solutions in the sense of distributions,
since the product of the discontinuous vector field $b$ with a singular
measure $f$ may not be well-defined. Instead, we directly define solutions in
duality with the nonconservative equation.

######  Definition 3.2.

A map $f\in C([0,T],\mathcal{M}_{\mathrm{loc},\mathrm{w}})$ is called a
solution of (3.2) if, for all $t\in[0,T]$ and $g\in C_{c}({\mathbb{R}}^{d})$,

$\int g(x)f(t,dx)=\int g(\phi_{0,t}(x))f_{0}(dx).$

######  Remark 3.2.

For $g\in C_{c}({\mathbb{R}}^{d})$ and $t\in[0,T]$, $(s,x)\mapsto
g(\phi_{s,t}(x))$ is the solution of the transport equation (3.1) in
$[0,t]\times{\mathbb{R}}^{d}$ with terminal value $g$ at time $t$, and, hence,
$f$ is called the duality solution of (3.2). Equivalently, $f(t,\cdot)$ is the
pushforward by $\phi_{0,t}$ of the measure $f_{0}$. When $f_{0}$ is a
probability measure, this means that $f(t,\cdot)$ is the law at time $t$ of
the stochastic process $\phi_{0,t}(X_{0})$, where $X_{0}$ is a random variable
with law $f_{0}$.

######  Remark 3.3.

The notion of duality solution can be equivalently formulated in relation to
nonconservative equations with a right-hand side777The theory of viscosity
solutions of the terminal value problem for (3.15) can be formulated following
the theory of the previous subsection with little change., that is, for $g\in
L^{1}([0,T],C({\mathbb{R}}^{d}))$,

(3.15) $-\partial_{t}u+b(t,x)\cdot\nabla u=g(t,x)\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}.$

With this perspective, although the object $\operatorname{div}(bf)$ does not
make sense as a classical distribution, the equation can still be applied to
particular singular test functions, namely, solutions of equations like
(3.15). Then the pairing

(3.16)
$\int_{{\mathbb{R}}^{d}}u(T,x)f(T,dx)-\int_{{\mathbb{R}}^{d}}u(0,x)f_{0}(dx)+\int_{0}^{T}\int_{{\mathbb{R}}^{d}}\underbrace{\left[-\partial_{t}u(t,x)+b(t,x)\cdot\nabla
u(t,x)\right]}_{=g(t,x)}f(t,dx)=0$

has a sense, because the singular terms collapse into a continuous function,
which may be paired with $f(t,\cdot)$.

######  Theorem 3.8.

There exists a unique duality solution $f$ of (3.2). If, for $\varepsilon>0$,
$f^{\varepsilon}$ is the solution corresponding to $b^{\varepsilon}$ as in
(2.3), then, as $\varepsilon\to 0$, $f^{\varepsilon}$ converges weakly in the
sense of measures to $f$. If $1\leq p<\infty$,
$f_{0},g_{0}\in\mathcal{P}_{p}$, and $f$ and $g$ are the corresponding duality
solutions, then, for some $C>0$ depending on $p$ and the constants in (2.1),
$\mathcal{W}_{p}(f_{t},g_{t})\leq C\mathcal{W}_{p}(f_{0},g_{0})$.

###### Proof.

The existence and uniqueness of duality solutions is a direct consequence of
the definition. Moreover, the duality solution identity implies that, for any
$R>0$ and for some $C>0$ depending on the constants in (2.1),
$\left\|f(t,\cdot)\right\|_{TV(B_{R})}\leq\left\|f_{0}\right\|_{TV(B_{R+C})}$.
For $0\leq s<t\leq T$ and $g\in C_{c}({\mathbb{R}}^{d})$, we apply the duality
formula with the test function $g\circ\phi_{s,t}$ and obtain the identity

$\int_{{\mathbb{R}}^{d}}g(x)f(t,dx)=\int_{{\mathbb{R}}^{d}}g(\phi_{s,t}(x))f(s,dx).$

Then, by Lemma 2.2, for some modulus of continuity $\omega$ depending on the
modulus of continuity for $g$,

$\left|\int_{{\mathbb{R}}^{d}}g(x)\left[f(t,dx)-f(s,dx)\right]\right|\leq\omega(|t-s|)\left\|f_{0}\right\|_{B_{\operatorname{supp}g+C}},$

and we conclude that $f\in C([0,T],\mathcal{M}_{\mathrm{loc},\mathrm{w}})$.

For $R>0$, define $f_{0,R}:=f_{0}\mathbf{1}_{B_{R}}$, and denote by $f_{R}$
and $f^{\varepsilon}_{R}$ the duality solutions of (3.2) with respectively $b$
and $b^{\varepsilon}$ and initial condition $f_{0,R}$. It then suffices to
prove that, for fixed $R>0$ as $\varepsilon\to 0$,
$f^{\varepsilon}_{R}\rightharpoonup f_{R}$ in the sense of measures. Then, in
view of Lemma 2.2, for any $t\in[0,T]$ and $g\in C_{c}({\mathbb{R}}^{d})$ for
sufficiently large support,

$\int_{{\mathbb{R}}^{d}}g(x)f_{R}(t,dx)=\int_{B_{R}}g(\phi_{0,t}(x))f_{0}(dx)=\int_{{\mathbb{R}}^{d}}g(\phi_{0,t}(x))f_{0}(dx)=\int_{{\mathbb{R}}^{d}}g(x)f(t,dx),$

and similarly for $f^{\varepsilon}$.

Let then $g\in C_{c}({\mathbb{R}}^{d})$ and $t\in(0,T]$ be fixed, and assume
without loss of generality that $f_{0}$ has compact support in $B_{R}$ for
some $R>0$. Then, for $\varepsilon>0$,

$\int_{{\mathbb{R}}^{d}}g(x)f^{\varepsilon}(t,dx)=\int
g(\phi^{\varepsilon}_{0,t}(x))f_{0}(dx).$

so that $\left\|f^{\varepsilon}\right\|_{TV}\leq\left\|f_{0}\right\|_{TV}$.
Moreover, if $\operatorname{supp}g\subset{\mathbb{R}}^{d}\backslash B_{R+C}$
for some $C>0$ sufficiently large and independent of $\varepsilon>0$, again by
Lemma 2.2,

$\int_{{\mathbb{R}}^{d}}g(x)f^{\varepsilon}(t,dx)=0.$

We may then take a weakly convergent subsequence of $f^{\varepsilon}$, with
limit point $F\in L^{\infty}([0,T],\mathcal{M})$, and, sending $\varepsilon\to
0$, we obtain that $F$ satisfies the duality solution identity, and therefore
$F=f$.

Choose $h_{1},h_{2}\in C_{c}({\mathbb{R}}^{d})$ such that, for all
$x,y\in{\mathbb{R}}^{d}$, $h_{1}(x)+h_{2}(y)\leq|x-y|^{p}$. Then, if $\gamma$
is any coupling between $f_{0}$ and $g_{0}$, we compute, using the duality
identity and Lemma 2.2,

$\displaystyle\int h_{1}(x)f(t,dx)+\int h_{2}(y)g(t,dy)$
$\displaystyle=\iint\left(h_{1}(\phi_{0,t}(x))+h_{2}(\phi_{0,t}(y))\right)\gamma(dx,dy)$
$\displaystyle\leq C\iint|x-y|^{p}\gamma(dx,dy).$

Taking the infimum over such $\gamma$ and supremum over such $h_{1},h_{2}$,
and using the dual formulation of the $p$-Wasserstein distance, we arrive at
the estimate for the Wasserstein distances. ∎

######  Remark 3.4.

The final estimate can also be proved using the characterization of $f$ and
$g$ as laws of certain stochastic processes (see Remark 3.2) and the
characterization of the Wasserstein metric in terms of random variables.

We may repeat the above analysis for the second-order conservative equation
(3.2), the only difference being the lack of a finite speed of propagation.
Therefore, all measures are taken to have finite mass over ${\mathbb{R}}^{d}$.
Below, $\Phi_{t,0}$ is the stochastic flow satisfying (3.7).

######  Definition 3.3.

A map $f\in C([0,T],\mathcal{M}_{\mathrm{w}})$ is called a solution of (3.4)
if, for all $t\in[0,T]$ and $g\in C_{b}({\mathbb{R}}^{d})$,

$\int g(x)f(t,dx)=\int{\mathbb{E}}[g(\Phi_{t,0}(x))]f_{0}(dx).$

######  Remark 3.5.

Once again, such solutions are called duality solutions because
${\mathbb{E}}[g\circ\Phi_{t,0}]$ is the solution of (3.3) with terminal value
$g$ at time $t$. If $f_{0}$ is a probability measure, then $f(t,\cdot)$ is the
law of the stochastic process $\Phi_{t,0}(X_{0})$, where $X_{0}$ is a random
variable with law $f_{0}$, independent of the Wiener process $W$.

The following may be proved exactly as for Theorem 3.8, now invoking the
properties of the stochastic flow described by Lemma 2.5.

######  Theorem 3.9.

There exists a unique duality solution $f$ of (3.4). If, for $\varepsilon>0$,
$f^{\varepsilon}$ is the solution corresponding to $b^{\varepsilon}$ as in
(2.3), then, as $\varepsilon\to 0$, $f^{\varepsilon}$ converges weakly in the
sense of measures to $f$. If $1\leq p\leq\infty$,
$f_{0},g_{0}\in\mathcal{P}_{p}$, and $f$ and $g$ are the corresponding duality
solutions, then, for some $C>0$ depending on $p$ and the constants in (2.1),
$\mathcal{W}_{p}(f_{t},g_{t})\leq C\mathcal{W}_{p}(f_{0},g_{0})$.

####  3.2.2 On the failure of renormalization

In view of the formula (3.6), it is immediate that (viscosity) solutions of
(3.1) satisfy the renormalization property, that is, if $u$ is a viscosity
solution and $\beta:{\mathbb{R}}\to{\mathbb{R}}$ is smooth, then $\beta\circ
u$ is also a solution. This is related to the existence and uniqueness of the
Lipschitz backward flow; indeed, note that, coordinate by coordinate,
$\phi_{t,T}(x)$ is the unique viscosity solution of (3.1) with terminal value
$x$ at time $T$.

We contrast this with the renormalization property for the forward,
conservative problem (3.2). If $b$ is smooth, then classical computations show
that $f$ is a solution if and only if $|f|$, $f_{+}$, and $f_{-}$ are all
solutions. Because $f(t,\cdot)$ is the pushforward by $f_{0}$ of the flow
$\phi_{0,t}$, this can be viewed as a generalized form of injectivity for the
flow. For general $b$ satisfying (2.1), the backward flow is not only not
injective, but concentrates at null sets. We therefore cannot expect
renormalization to hold in general.

As a concrete example, take again $b(x)=\operatorname{sgn}x$ on
${\mathbb{R}}$, and $f_{0}=\frac{1}{2}\delta_{1}-\frac{1}{2}\delta_{-1}$.
Then, for $t>0$,
$f(t,\cdot)=\frac{1}{2}\delta_{(1-t)_{+}}-\frac{1}{2}\delta_{-(1-t)_{+}}$,
which means that $f(t,\cdot)\equiv 0$ for $t\geq 1$. However, the solution $F$
of (3.2) with $F_{0}=|f_{0}|=\frac{1}{2}\delta_{1}+\frac{1}{2}\delta_{-1}$ is
equal to
$F(t,\cdot)=\frac{1}{2}\delta_{(1-t)_{+}}+\frac{1}{2}\delta_{-(1-t)_{+}}$, so
that $F(t,\cdot)=\delta_{0}$ for $t\geq 1$. Thus $F_{t}\neq|f_{t}|$ for $t\geq
1$; indeed, $|f_{t}|$ does not even conserve mass.

The failure of renormalization holds even if we impose $f_{0}\in L^{1}\cap
L^{\infty}$. For such $f_{0}$ and for $b(x)=\operatorname{sgn}x$, we have

$f(t,dx)=\left[f_{0}(x+t)\mathbf{1}\left\\{x>0\right\\}+f_{0}(x-t)\mathbf{1}\left\\{x<0\right\\}\right]dx+\left(\int_{[-t,t]}f_{0}\right)d\delta_{0}(x).$

Therefore, renormalization fails whenever $f_{0}$ is nonzero and odd.

We present one more counterexample to renormalization in which $b\in C$ and
$f\in L^{1}$ (as the previous example shows, even if $f_{0}\in L^{1}$,
$f(t,\cdot)$ may not be absolutely continuous with respect to Lebesgue measure
due to the concentration of the flow). Take
$b(t,x)=2\operatorname{sgn}x|x|^{1/2}$. The backward flow is given by
$\phi_{0,t}(x)=\operatorname{sgn}x(|x|^{1/2}-t)_{+}^{2}$ for
$(t,x)\in[0,T]\times{\mathbb{R}}$. For $f_{0}\in L^{1}$, the duality solution
is given by

$f(t,dx)=\left(\int_{[-t^{2},t^{2}]}f_{0}\right)\delta_{0}(dx)+f_{0}\left(\operatorname{sgn}x(|x|^{1/2}+t)^{2}\right)\frac{|x|^{1/2}+t}{|x|^{1/2}}dx.$

We then take the odd density
$f_{0}(x)=\operatorname{sgn}x|x|^{1/2}\mathbf{1}_{[-1,1]}(x)$, and the duality
solution takes values in $L^{1}$:

(3.17)
$f(t,x)=\operatorname{sgn}x\frac{(|x|^{1/2}+t)^{2}}{|x|^{1/2}}\mathbf{1}_{[-(1-t)_{+}^{2},(1-t)_{+}^{2}]}(x).$

On the other hand, $|f|$ is not the duality solution, or even a distributional
solution, since mass is not conserved. The unique duality solution with
initial density $|f_{0}(x)|=|x|^{1/2}\mathbf{1}_{[-1,1]}(x)$ in this case is
given by

$F(t,dx)=\frac{4t^{3}}{3}\delta_{0}(dx)+\frac{(|x|^{1/2}+t)^{2}}{|x|^{1/2}}\mathbf{1}_{[-(1-t)_{+}^{2},(1-t)_{+}^{2}]}(x)dx.$

######  Remark 3.6.

One consequence of the commutator lemma of DiPerna and Lions [37, Lemma II.1]
is that, if $f\in L^{p}$ and $b\in W^{1,q}$ with $\frac{1}{p}+\frac{1}{q}\leq
1$, then the renormalization property is satisfied. The previous example
therefore indicates that these conditions cannot be weakened in general.
Indeed, even though $f_{0}\in L^{1}\cap L^{\infty}$, the solution $f(t,\cdot)$
given by (3.17) belongs to $L^{p}$ only for $p\in[1,2)$ when $t>0$, and the
same is true for $\partial_{x}b$.

####  3.2.3 Equivalence of duality and distributional solutions

We finish this section by studying the setting where $bf$ can be understood as
a distribution, and, therefore, distributional solutions of (3.2) can be
considered.

######  Theorem 3.10.

Assume either that $b$ is continuous, or that $f(t,\cdot)\in
L^{1}_{\mathrm{loc}}$ for all $t\in[0,T]$. Then $f$ is a distributional
solution of (3.2) if and only if $f$ is the unique duality solution.

###### Proof.

Suppose $f$ is the unique duality solution. Let
$(b^{\varepsilon})_{\varepsilon>0}$ be as in (2.3) and let $f^{\varepsilon}$
be the corresponding solution of (3.2). For $\phi\in
C^{1}_{c}((0,T)\times{\mathbb{R}}^{d})$, integrating by parts yields

$\iint_{(0,T)\times{\mathbb{R}}^{d}}f^{\varepsilon}(t,x)\left(-\partial_{t}\phi(t,x)+b^{\varepsilon}(t,x)\cdot\nabla\phi(t,x)\right)dtdx=0.$

In the case that $b\in C$, we may choose regularizations $b^{\varepsilon}$
that converge locally uniformly to $b$. By Theorem 3.8, as $\varepsilon\to 0$,
$f^{\varepsilon}$ converges weakly in the sense of measures to $f$, and so we
may take $\varepsilon\to 0$ above to obtain

$\iint_{(0,T)\times{\mathbb{R}}^{d}}f(t,dx)\left(-\partial_{t}\phi(t,x)+b(t,x)\cdot\nabla\phi(t,x)\right)dt=0.$

Otherwise, if $f\in L^{1}_{\mathrm{loc}}$, it follows that $f^{\varepsilon}$
converges weakly in $L^{1}_{\mathrm{loc}}$, and therefore the same is true for
$b^{\varepsilon}f^{\varepsilon}$ by the dominated convergence theorem. We may
then take $\varepsilon\to 0$ in this case as well.

Assume now that $f$ is an arbitrary distributional solution. We aim to show
the duality equality in Definition 3.2, and, by a density argument, it
suffices to do so for $g\in C_{c}({\mathbb{R}}^{d})\cap
C^{0,1}({\mathbb{R}}^{d})$. Let $\rho_{\varepsilon}$ be a standard mollifier
as before and set $f_{\varepsilon}=f*\rho_{\varepsilon}$. Then
$f_{\varepsilon}$ satisfies

$\partial_{t}f_{\varepsilon}-\operatorname{div}(bf_{\varepsilon})=\operatorname{div}r_{\varepsilon},$

where $r_{\varepsilon}=(bf)*\rho_{\varepsilon}-bf_{\varepsilon}$. For
$t\in(0,T]$, let $u$ be the unique Lipschitz viscosity solution of the
terminal value problem

$-\partial_{s}u+b\cdot\nabla u=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad u(t,\cdot)=g.$

By the theory in subsection 3.1, $u(s,x)=g(\phi_{s,t}(x))$ and is Lipschitz
continuous with compact support. We then compute

$\partial_{s}\int f_{\varepsilon}(s,x)u(s,x)dx=-\int
r_{\varepsilon}(s,x)\cdot\nabla u(s,x)dx,$

so that

$\int
f_{\varepsilon}(t,x)g(x)dx-\int(f_{0}*\rho_{\varepsilon})(x)g(\phi_{0,t}(x))dx=-\int_{0}^{t}\int_{{\mathbb{R}}^{d}}r_{\varepsilon}(s,x)\cdot\nabla
u(s,x)dxds.$

We may then conclude by proving that
$r_{\varepsilon}\xrightarrow{\varepsilon\to 0}0$ in $L^{1}_{\mathrm{loc}}$.

If $f\in L^{1}_{\mathrm{loc}}$, this is immediate because, as $\varepsilon\to
0$, both $(bf)*\rho_{\varepsilon}$ and $bf_{\varepsilon}$ converge in
$L^{1}_{\mathrm{loc}}$ to $bf$. If $b\in C$, then, as $\varepsilon\to 0$, both
$(bf)*\rho_{\varepsilon}$ and $bf_{\varepsilon}$ converge locally in total
variation to $bf$. It follows that $r_{\varepsilon}$ converges locally in
total variation to $0$, but, because $r_{\varepsilon}\in L^{1}$ for all
$\varepsilon>0$, the convergence in $L^{1}_{\mathrm{loc}}$ is established. ∎

######  Remark 3.7.

Even in the context of Theorem 3.10, the renormalization property can fail.
Indeed, this is the case for the final example in the previous subsubsection,
where both $b\in C$ and $f\in L^{1}$.

##  4 The expansive regime

We continue our analysis of transport and continuity equations with vector
fields $b$ satisfying (2.1), and in this section we study the expansive
regime. Reversing the sign appearing in front of the velocity field $b$, the
initial value problem for the continuity equation becomes

(4.1) $\partial_{t}f+\operatorname{div}(b(t,x)f)=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}\quad\text{and}\quad f(0,\cdot)=f_{0},$

and the corresponding dual terminal value problem for the non-conservative
transport equation is

(4.2) $\partial_{t}u+b(t,x)\cdot\nabla u=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d}\quad\text{and}\quad u(T,\cdot)=u_{T}.$

Equivalently, we are studying the time-reversed versions of (3.1) and (3.2)
(in this case, $b$ is replaced with $b(T-t,\cdot)$). As such, the relevant
direction of the flow (2.2) changes in this context: whereas in the previous
section, the compressive, backward flow gave rise to the dual solution spaces
$C$ and $\mathcal{M}$, here, the expansive, forward flow allows to develop a
theory for both (4.1) and (4.2) in Lebesgue spaces. This can also be seen from
formal a priori $L^{p}$ estimates for (4.1) and (4.2), which follow
immediately from the lower bound on $\operatorname{div}b$.

The regime for these equations matches those studied by Bouchut, James, and
Mancini [23], in which emphasis is placed on the fact that distributional
solutions $f\in
C([0,T],L^{\infty}_{{\mathrm{w}\text{-}\star}}({\mathbb{R}}^{d}))$ of (4.1)
are not unique in general. Our approach to these equations is similar, in that
we use a particular solution of (4.1) to study, by duality, the transport
equation (4.2) and the forward ODE flow to (2.2). We extend the results of
[23] by identifying a “good” solution of (4.2) for any $f_{0}\in
L^{p}_{\mathrm{loc}}$, where the continuous solution operator on $L^{p}$ is
stable under regularizations in the weak topology of
$C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$.

The terminal value problem (4.2) is then understood both in the dual sense and
through the lens of renormalization theory. It is this theory that allows, as
in [37], to make sense of the forward ODE flow (2.17) as the right-inverse of
the backward flow, completing the program initiated in Section 2. As a
consequence, we then also obtain the uniqueness of nonnegative distributional
solutions of (4.1), and, by extension, a characterization of the “good”
solution.

We finish the section by making some remarks about the second-order analogues
of (4.1) and (4.2). Unlike in the previous section, we do not have a full
solution theory for general second-order equations, unless the ellipticity
matrix is uniformly positive (the case which has already been covered by
Figalli in [38]) or is degenerate but independent of the space variable.

###  4.1 The conservative equation

The starting point for the study of the conservative equation (4.1) is that
distributional solutions in the sense of distributions are not unique (see
also [20], [23, Section 6]). We revisit the example, when $d=1$,
$b(t,x)=\operatorname{sgn}x$. Then
$f(t,x):=\operatorname{sgn}x\mathbf{1}_{|x|\leq t}$ is a nontrivial
distributional solution of (4.1) belonging to $L^{1}\cap L^{\infty}$ with
$f(0,\cdot)=0$. The uniqueness can be seen as a consequence of the contractive
nature of the backward flow (2.2), which allows for positive and negative mass
to be “cancelled” at time $0$, only to appear immediately for $t>0$. The same
phenomenon is what leads to the failure of renormalization for the contractive
regime for the continuity equation in subsection 3.2. In either case, we
remark that, this particular $b$ belongs to $BV({\mathbb{R}})$, while
$\partial_{x}b$ is not absolutely continuous with respect to Lebesgue measure,
and so the condition in the work of Ambrosio [4] that $\operatorname{div}b\in
L^{1}_{\mathrm{loc}}$ cannot indeed be weakened in general, if one is to hope
for renormalization or uniqueness for the continuity equation.

One strategy is to define solutions of (4.1) by duality with the transport
equation (3.1) from the contractive setting. With the theory of Section 3, for
$g\in C^{0,1}_{c}({\mathbb{R}}^{d})$, we may define a Lipschitz viscosity
solution of the initial value problem

$\partial_{t}v+b(t,x)\cdot\nabla v=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad v(0,\cdot)=g$

(because $\tilde{v}(t,x):=v(T-t,x)$ solves the corresponding terminal value
problem (3.1) with velocity $\tilde{b}(t,x)=b(T-t,x)$), and then, formally,
for $t>0$, $\int f(t,x)v(t,x)dx=\int f_{0}(x)g(x)dx$.

The main problem with this approach is that duality does not define unique
solutions, again due to the concentration effect of the backward flow. Taking
once more $b(t,x)=\operatorname{sgn}x$, we have, by (3.6),

$v(t,x)=\begin{dcases}g(x-(\operatorname{sgn}x)t),&|x|\geq t,\\\ g(0),&|x|\leq
t.\end{dcases}$

Therefore, the duality equality fails to give sufficient information to
identify $f$ in the cone $\\{|x|\leq t\\}$, in which $v$ is always constant,
regardless of the initial data $g$. Indeed, the two distributional solutions
$f\equiv 0$ and $f(t,x)=\operatorname{sgn}x\mathbf{1}\\{|x|\leq t\\}$ differ
in exactly this cone, in which the Jacobian of the backward flow vanishes. It
is exactly this observation that lead to the notion of “exceptional” solutions
of (3.1) and the exceptional set in [23].

We instead identify a “good” solution operator acting on all $f_{0}\in
L^{p}_{\mathrm{loc}}$, $1\leq p\leq\infty$, by extending the solution formula
in the smooth case, which depends on the backward flow studied in Section 2,
as well as the corresponding Jacobian. In particular, the “good solution” is
distinguished by vanishing whenever the Jacobian does. Our approach differs
slightly from that of [23], who work with a general class of “transport flows”
that generalize the backward ODE flow. One advantage of our analysis is that
we can directly appeal to the various topological properties of the backward
flow proved in Section 2.

####  4.1.1 Representation formula

If $b$ is Lipschitz, then the solution of (4.1) is given by

(4.3) $f(t,x)=f_{0}(\phi_{0,t}(x))J_{0,t}(x),$

where $\phi_{0,t}(x)$ is the reverse flow defined in Section 2 and
$J_{0,t}(x)=\det(\nabla_{x}\phi_{0,t}(x))$ is the corresponding Jacobian. One
way to derive this formula is through the Feynman-Kac formula for the reversed
time equation

$-\partial_{t}\tilde{f}+b(T-t,x)\cdot\nabla\tilde{f}+\operatorname{div}_{x}b(T-t,x)\tilde{f}=0\quad\text{in
}(0,T)\times{\mathbb{R}}^{d},\quad\tilde{f}(T,\cdot)=f_{0},$

which gives

(4.4)
$f(t,x)=\tilde{f}(T-t,x)=f_{0}(\phi_{0,t}(x))\exp\left(-\int_{0}^{t}\operatorname{div}b(s,\phi_{0,s}(x)ds\right),$

and then
$J_{0,t}(x)=\exp\left(-\int_{0}^{s}\operatorname{div}b(s,\phi_{0,s}(x)ds\right)$.

In the general case where $b$ satisfies (2.1), the formula (4.3) makes sense
for arbitrary $f_{0}\in L^{p}_{\mathrm{loc}}$, $1\leq p\leq\infty$. We may
then use the various results in Section 2 to analyze the stability properties
of the solution operator defined by the formula (3.6). We remark in particular
that the stability results of Lemma 2.3 depend on the determinant structure of
the Jacobian, which is somewhat disguised by the exponential expression in
(4.4).

######  Theorem 4.1.

Let $1\leq p\leq\infty$, assume that $f_{0}\in
L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})$, and define $f$ by (4.3). Then $f$ is
a distributional solution of (4.1). If $1\leq p<\infty$, $f\in
C([0,T],L^{p}({\mathbb{R}}^{d}))$, and if $p=\infty$, $f\in
C([0,T],L^{\infty}_{\mathrm{w}\text{-}\star}({\mathbb{R}}^{d}))$. There exists
a constant $C>0$ depending only on the assumptions in (2.1) such that, for all
$R>0$,

(4.5) $\left\|f(t,\cdot)\right\|_{L^{p}(B_{R})}\leq
C\left\|f_{0}\right\|_{L^{p}(B_{R+C})}.$

If $(b_{\varepsilon})_{\varepsilon>0}$ are as in (2.3) and
$(f_{\varepsilon})_{\varepsilon>0}$ are the corresponding solutions of (4.1),
then, as $\varepsilon\to 0$, $f_{\varepsilon}$ converges to $f$ weakly in
$C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ if $1\leq p<\infty$, and
weak-$\star$ in $L^{\infty}$ if $p=\infty$.

###### Proof.

When $p=\infty$, the bound (4.5) follows from the $L^{\infty}$ bounds for the
flow and Jacobian in Lemmas 2.2 and 2.3. We prove the bound when $p<\infty$
for the solutions $f_{\varepsilon}$ of the equation with $b_{\varepsilon}$ as
in (2.3), for a constant independent of $\varepsilon$, and then the estimate
for $f$ follows after proving the weak convergence result.

For a constant $C>0$ independent of $\varepsilon$, by Lemmas 2.2 and 2.3, we
have $|J_{0,t}|\leq C$ and $|\phi_{0,t}(x)|\leq R+C$ for $|x|\leq R$. Then

$\displaystyle\int_{B_{R}}|f^{\varepsilon}(t,x)|^{p}dx$
$\displaystyle=\int_{{\mathbb{R}}^{d}}|f_{0}(\phi^{\varepsilon}_{0,t}(x))|^{p}J^{\varepsilon}_{0,t}(x)^{p}dx$
$\displaystyle\leq\left\|J_{0,t}\right\|_{\infty}^{p-1}\int_{B_{R}}|f_{0}(\phi^{\varepsilon}_{0,t}(x))|^{p}J^{\varepsilon}_{0,t}(x)dx\leq
C\int_{B_{R+C}}|f_{0}(x)|^{p}dx.$

It suffices to prove the weak convergence of $f^{\varepsilon}$ when $p<\infty$
for $f_{0}\in C_{c}$. In the general case, if $\tilde{f}_{0}$ is continuous
with compact support and we let $\tilde{f}^{\varepsilon}$ be the solution with
$b^{\varepsilon}$ and $\tilde{f}_{0}$, we have

$\left\|f^{\varepsilon}-\tilde{f}^{\varepsilon}\right\|_{C([0,T],L^{p}(B_{R}))}\leq
C\left\|f_{0}-\tilde{f}_{0}\right\|_{L^{p}(B_{R+C})},$

and we may then choose $\tilde{f}_{0}$ arbitrarily close to $f_{0}$ in
$L^{p}_{\mathrm{loc}}$.

By Lemma 2.2, as $\varepsilon\to 0$, $\phi^{\varepsilon}\to\phi$ uniformly in
$[0,T]\times{\mathbb{R}}^{d}$, and therefore
$f_{0}\circ\phi^{\varepsilon}_{0,t})$ converges uniformly to
$f_{0}\circ\phi_{0,t}$. In view of Lemma 2.3, $f^{\varepsilon}$ converges
weakly in the sense of distributions (and therefore, in the sense of locally
bounded Borel measures) to $f$. Since $f^{\varepsilon}$ is bounded in
$L^{\infty}([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$, the convergence is
actually weak in $L^{\infty}([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$.

If $p=\infty$, then, in particular, $f^{\varepsilon}\in
C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ for $p<\infty$, uniformly in
$\varepsilon$, and we have the convergence as $\varepsilon\to 0$ in the sense
of distributions to $f$. In this case, $f\in
L^{\infty}_{\mathrm{loc}}([0,T]\times{\mathbb{R}}^{d})$, and so the
convergence is weak-$\star$ in $L^{\infty}_{\mathrm{loc}}$.

Given $g\in C^{1}_{c}((0,T)\times{\mathbb{R}}^{d})$, integrating by parts
gives

$\iint_{[0,T]\times{\mathbb{R}}^{d}}f^{\varepsilon}(t,x)\left[\partial_{t}\phi(t,x)+b^{\varepsilon}(t,x)\cdot\nabla\phi(t,x)\right]dxdt=0.$

As $\varepsilon\to 0$, the bracketed expression converges a.e. to
$\partial_{t}\phi(t,x)+b(t,x)\cdot\nabla\phi(t,x)$, and so converges weakly in
$L^{q}$ for all $1\leq q<\infty$ by the dominated convergence theorem. We may
therefore send $\varepsilon\to 0$, using the weak convergence of
$f^{\varepsilon}$, to deduce that $f$ is a distributional solution. This
implies in particular that $f\in
C([0,T],L^{p}_{\mathrm{w}}({\mathbb{R}}^{d}))$, or
$C([0,T],L^{\infty}_{\mathrm{w}\text{-}\star}({\mathbb{R}}^{d}))$ if
$p=\infty$.

To show that $f\in C([0,T],L^{p}({\mathbb{R}}^{d}))$ when $p<\infty$, we may
again consider $f_{0}\in C_{c}({\mathbb{R}}^{d}))$ without loss of generality.
Then $f_{0}\circ\phi_{0,\cdot}\in C([0,T]\times{\mathbb{R}}^{d})$, while
$J_{0,\cdot}\in C([0,T],L^{1}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ by Lemma 2.3,
and the result follows. ∎

######  Remark 4.1.

We often call the solution $f$ defined through (4.3) the “good” solution. In
view of the stability results of Theorem 4.1 above, this solution coincides
with the notion of reversible solutions in [20, 23].

The following is immediate from the formula (4.3).

######  Corollary 4.1.

If $f$ is a “good” solution of (4.1), then so is $|f|$.

Corollary 4.1 is in direct contrast to the continuity equation in the
compressive setting of the previous section, where renormalization fails. Its
proof depends on the formula for the good solution; indeed, despite the weak
stability result in Theorem 4.1, this renormalization property cannot be
proved by regularization, since we only have the weak convergence as
$\varepsilon\to 0$ of $f_{\varepsilon}$ to $f$. At present, we do not know
whether the convergence is strong in $L^{p}$. This turns out to be equivalent
to the strong convergence in $L^{1}_{\mathrm{loc}}$ of the Jacobians, and
therefore, in view of Proposition 2.1, we have the following when $d=1$.

######  Theorem 4.2.

Assume $d=1$, $f_{0}\in L^{p}_{\mathrm{loc}}({\mathbb{R}})$ for $p<\infty$,
$(b^{\varepsilon})_{\varepsilon>0}$ is as in (2.3), and $f^{\varepsilon}$ is
the corresponding solution of (4.1). Then, as $\varepsilon\to 0$,
$f^{\varepsilon}$ converges strongly in
$C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}))$ to $f$.

###### Proof.

Just as in the proof of Theorem 4.1, we may assume without loss of generality
that $f_{0}\in C_{c}({\mathbb{R}})$. In that case, $f^{\varepsilon}$ is
bounded in $L^{1}$ and $L^{\infty}$, and so the strong $L^{p}$ convergence
reduces to the strong convergence of $J^{\varepsilon}_{0,\cdot}$ to
$J_{0,\cdot}$ in $L^{1}_{\mathrm{loc}}([0,T]\times{\mathbb{R}})$ from
Proposition 2.1. ∎

####  4.1.2 Vanishing viscosity approximation

The good solution above also arises from vanishing viscosity limits, that is,
the limit as $\varepsilon\to 0$ of solutions of

(4.6) $\partial_{t}f^{\varepsilon}-\frac{\varepsilon^{2}}{2}\Delta
f^{\varepsilon}+\operatorname{div}(b(t,x)f^{\varepsilon})=0\quad\text{in
}[0,T]\times{\mathbb{R}}^{d}\quad\text{and}\quad
f^{\varepsilon}(0,\cdot)=f_{0},$

which has as its unique solution

(4.7)
$f^{\varepsilon}(t,x):={\mathbb{E}}[f_{0}(\phi^{\varepsilon}_{0,t}(x))J^{\varepsilon}_{0,t}(x)],$

where now $\phi^{\varepsilon}$ and $J^{\varepsilon}$ denote respectively the
stochastic flow and Jacobian from (2.23), corresponding to Proposition 2.3.

The proof of the following result follows from Proposition 2.3, and is proved
almost exactly as for Theorem 4.1.

######  Theorem 4.3.

The function $f^{\varepsilon}$ defined by (4.6) belongs to
$C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ if $1\leq p<\infty$ and
$C([0,T],L^{\infty}_{\mathrm{w}\text{-}\star}({\mathbb{R}}^{d}))$ if
$p=\infty$, and, as $\varepsilon\to 0$, $f^{\varepsilon}$ converges weakly in
those spaces to $f$.

###  4.2 The nonconservative equation

The next step is the study of the terminal value problem (4.2). Unlike the
transport equation (3.1) with velocity $-b$, which was solved in the space of
continuous functions, we cannot define $L^{p}$ solutions in the distributional
sense, as the product $b\cdot\nabla
u=\operatorname{div}(bu)-(\operatorname{div}b)u$ does not make sense when
$\operatorname{div}b$ is merely a measure. Instead, we initially characterize
solutions by duality with (4.1), which can be seen as a way of restricting the
class of test functions to deal with the singularities in $b$ (see Remark
3.3).

####  4.2.1 $L^{p}$ and $BV$ estimates

We will first prove a priori $L^{p}$ and $BV$ estimates for the solution of
(4.2), assuming all the data and solutions are smooth. The $BV$ estimates in
particular are crucial to establishing the strong convergence in $L^{p}$ of
regularized solutions to a unique limit, which will be the duality solution,
adjoint to the equation (4.1). The $BV$ estimate appears already in [23, Lemma
4.4]. We present an alternate proof here, which is similar to the one for
second-order equations we prove later.

######  Lemma 4.1.

Assume $b$ is smooth and satisfies (2.1), and let $u$ be a smooth solution of
(4.2). Then, for all $1\leq p\leq\infty$, there exist $C=C_{p,R}\in
L^{1}_{+}([0,T])$ and $C_{R}>0$ depending only on the bounds in (2.1) such
that, for all $0\leq t\leq T$,

$\left\|u(t,\cdot)\right\|_{L^{p}(B_{R})}\leq\exp\left(\int_{0}^{t}C(s)ds\right)\left\|u_{T}\right\|_{L^{p}(B_{R+C})}\quad$

and

$\left\|u(t,\cdot)\right\|_{BV(B_{R})}\leq\exp\left(\int_{0}^{t}C(s)ds\right)\left\|u_{T}\right\|_{BV(B_{R+C})}.$

###### Proof.

We assume that $u_{T}$ has compact support, and, therefore, in view of the
finite speed of propagation property, so does $u$. The general result for
$L^{p}_{\mathrm{loc}}$ and $BV_{\mathrm{loc}}$ is proved similarly.

The $L^{\infty}$ bound is a consequence of the maximum principle. For
$p<\infty$, we compute

$\frac{\partial}{\partial
t}\int_{{\mathbb{R}}^{d}}|u(t,x)|^{p}dx=\int_{{\mathbb{R}}^{d}}\operatorname{div}b(t,x)|u(t,x)|^{p}dx\geq-
C_{0}(t)d\int_{{\mathbb{R}}^{d}}|u(t,x)|^{p}dx,$

and the $L^{p}$ bound follows from Grönwall’s inequality.

Now, for $t\leq T$ and $x,z\in{\mathbb{R}}^{d}$, set $w(t,x,z)=\nabla
u(t,x)\cdot z$. Then $w$ satisfies

$-\partial_{t}w+b\cdot\nabla_{x}w+(z\cdot\nabla)b\cdot\nabla_{z}w=0.$

Since $b$ and $w$ are smooth, the renormalization property holds for this
transport equation, and so a simple regularization argument shows, in the
sense of distributions,

$\partial_{t}|w|+b\cdot\nabla_{x}|w|+(z\cdot\nabla)b\cdot\nabla_{z}|w|=0.$

Define $\phi(z)=e^{-|z|^{2}}$. Then

$\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\phi(z)b(t,x)\cdot\nabla_{x}|w(t,x,z)|dxdz=-\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\phi(z)\operatorname{div}b(t,x)|w(t,x,z)|dxdz$

and

$\displaystyle\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\phi(z)(z\cdot\nabla)b(t,x)\cdot\nabla_{z}|w(t,x,z)|dxdz$
$\displaystyle=-\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\left[\nabla\phi(z)\cdot(z\cdot\nabla
b(t,x))+\phi(z)\operatorname{div}b(t,x)\right]|w(t,x,z)|dxdz.$

Therefore, by Lemma 2.1,

$\displaystyle\partial_{t}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|w(t,x,z)|\phi(z)dxdz$
$\displaystyle=\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\left[\nabla\phi(z)\cdot(z\cdot\nabla
b(t,x))+2\phi(z)\operatorname{div}b(t,x)\right]|w(t,x,z)|dxdz$
$\displaystyle=\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}2e^{-|z|^{2}}\left[\operatorname{div}b(t,x)-\nabla
b(t,x)z\cdot z\right]dxdz$
$\displaystyle\geq-2(d-1)C_{0}(t)\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}e^{-|z|^{2}}|w(t,x,z)|dxdz.$

The result follows from Grönwall’s lemma and the fact that

$\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}e^{-|z|^{2}}|w(t,x,z)|dxdz=c_{0}\int_{{\mathbb{R}}^{d}}|\nabla
u(t,x)|dx,$

where the constant $c_{0}=\int_{{\mathbb{R}}^{d}}e^{-|z|^{2}}|\nu\cdot z|dz$
is independent of the choice of $|\nu|=1$ by rotational invariance. ∎

####  4.2.2 Duality solutions

Proceeding by duality with the conservative forward equation, and using the
$BV$-estimates above, then gives the following.

######  Theorem 4.4.

Assume $1\leq p\leq\infty$ and $u_{T}\in L^{p}_{\mathrm{loc}}$. Then there
exists a unique function $u\in
C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ (or in
$C([0,T],L^{\infty}_{\mathrm{w}\text{-}\star}({\mathbb{R}}^{d}))$ if
$p=\infty$) such that, if $(b^{\varepsilon})_{\varepsilon>0}$ is as in (2.3)
and $u^{\varepsilon}$ denotes the corresponding solution of (4.2), then, as
$\varepsilon\to 0$, $u^{\varepsilon}$ converges strongly in
$C([0,T],L^{p}({\mathbb{R}}^{d}))$ for $p<\infty$ and weak-$\star$ in
$L^{\infty}$ to $u$. Moreover, the solution map $u_{T}\mapsto u$ is linear,
order-preserving, and continuous in $L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$.
If $s\in[0,T)$, $f_{s}\in L^{p^{\prime}}({\mathbb{R}}^{d})$ and $f\in
C([s,T],L^{p^{\prime}}({\mathbb{R}}^{d}))$ (or
$C([s,T],L^{\infty}_{\mathrm{w}\text{-}\star}({\mathbb{R}}^{d}))$ if $p=1$) is
the good solution of (4.1) with initial data $f(s,\cdot)=f_{s}$, then

$\int_{{\mathbb{R}}^{d}}u(s,x)f_{s}(x)dx=\int_{{\mathbb{R}}^{d}}u_{T}(x)f(T,x)dx.$

######  Remark 4.2.

The function $u$ corresponds with the notion of duality solution presented in
[23] whenever $u_{T}$ (and therefore $u(t,\cdot)$ for $t<T$) belongs to
$BV_{\mathrm{loc}}$.

###### Proof.

By Lemma 4.1, $(u^{\varepsilon})_{\varepsilon>0}$ is bounded uniformly in
$C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$, and so, along a
subsequence, converges weakly as $\varepsilon\to 0$ to some $u$ satisfying the
same bounds.

In order to see that the convergence is strong, note that it suffices, by the
$L^{p}$-boundedness of solution operator implied by Lemma 4.1, to assume that
$u_{T}\in C_{c}({\mathbb{R}}^{d})$. We then have $u^{\varepsilon}$ bounded in
$L^{\infty}([0,T],BV({\mathbb{R}}^{d}))$ independently of $\varepsilon$. The
identity $\partial_{t}u^{\varepsilon}=-b^{\varepsilon}\cdot\nabla
u^{\varepsilon}$ then implies that, for any $t_{1}<t_{2}\leq T$ and $R>0$,

$\left\|u^{\varepsilon}(t_{1},\cdot)-u^{\varepsilon}(t_{2},\cdot)\right\|_{L^{1}(B_{R})}\leq\left\|b\right\|_{L^{\infty}(B_{R})}\sup_{t\in[0,T]}\left\|\nabla
u^{\varepsilon}\right\|_{L^{1}(B_{R})}|t_{1}-t_{2}|.$

This, along with the uniform $BV$ estimates, implies that
$(u^{\varepsilon})_{\varepsilon>0}$ is precompact in
$C([0,T],L^{1}_{\mathrm{loc}}({\mathbb{R}}^{d}))$, and, because of the uniform
$L^{\infty}$-bound, precompact in
$C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ for any $p\in[1,\infty)$. It
therefore follows that any weakly convergent subsequence actually converges
strongly.

If $f^{\varepsilon}$ is the solution of (4.1) with
$f^{\varepsilon}(s,\cdot)=f_{s}$, then classical computations involving
integration by parts give

$\int_{{\mathbb{R}}^{d}}u^{\varepsilon}(s,x)f_{s}(x)dx=\int_{{\mathbb{R}}^{d}}u_{T}(x)f^{\varepsilon}(T,x)dx.$

Sending $\varepsilon\to 0$ along a subsequence and using the weak convergence
of $f^{\varepsilon}$ and strong convergence of $u^{\varepsilon}$ shows that
any limit point $u$ must satisfy the duality identity with $f$, and is
therefore unique. We conclude that the full sequence converges strongly. As
before, when $p=\infty$, we obtain the same result since then also $u\in
C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ for any $p<\infty$.

∎

######  Remark 4.3.

If $u_{T}\in BV_{\mathrm{loc}}$, then the duality solution $u$ of (4.2)
satisfies $\nabla u\in
L^{\infty}([0,T],\mathcal{M}_{\mathrm{loc}}({\mathbb{R}}^{d}))$. Note,
however, that this is still not enough to make sense of $u$ as a
distributional solution, unless $b$ is continuous.

####  4.2.3 Renormalization

In Section 3, the renormalization property for solutions of the transport
equation (3.1) followed from the formula (3.6). We prove a similar
renormalization property for the transport equation (4.2) in the expansive
regime. Here, it depends on the strong convergence in $L^{p}$ of
regularizations.

######  Theorem 4.5.

Let $1\leq p\leq\infty$ and $u_{T}\in L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})$,
and let $u\in C([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ be the duality
solution of (4.2). Assume $\beta:{\mathbb{R}}\to{\mathbb{R}}$ is smooth and
satisfies $|\beta(r)|\leq(1+|r|^{\alpha})$ for some $\alpha>0$. Then
$\beta\circ u=C([0,T],L^{p/\alpha}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ is the
duality solution of (4.2) with terminal value $\beta(u(T,\cdot))=\beta\circ
u_{T}$.

###### Proof.

The proof is an easy consequence of regularization of $b$ as in (2.3), and the
passage to the limit follows from the strong convergence of $u^{\varepsilon}$
to $u$. ∎

###  4.3 The forward ODE flow

We finally return to the study of the flow (2.2), in particular for the
forward direction. A candidate for the object $\phi_{t,s}(x)$, $t>s$, a.e. $x$
was already identified in Proposition 2.2 as the right inverse of the backward
flow—note that the full measure set of $x\in{\mathbb{R}}^{d}$ depends on $s$
and $t$. We now connect this right-inverse with the transport equation (4.2),
and exploit the renormalization property to identify $\phi_{t,s}(x)$ as a
regular Lagrangian flow, that is, for a.e. $x\in{\mathbb{R}}^{d}$, an
absolutely continuous solution of the integral equation for (2.2) with control
on the compressibility.

####  4.3.1 Properties of the right inverse

We first record more properties of the right-inverse of the backward flow
identified in Proposition 2.2. From now on, for $0\leq s\leq t\leq T$, we
always denote by $\phi_{t,s}$ the version of the right-inverse of $\phi_{s,t}$
which is continuous almost everywhere (such a version is guaranteed to exist
by Proposition 2.2).

######  Theorem 4.6.

For any $t\in(0,T]$, $(s,x)\mapsto\phi_{s,t}(x)$ is (coordinate-by-coordinate)
the duality solution of (4.2) with terminal value $x$ at time $t$. For all
$1\leq p<\infty$,

$\phi_{\cdot,s}\in
C([s,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))\quad\text{and}\quad\phi_{t,\cdot}\in
C([0,t],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})),$

and there exists a constant $C>0$ such that, for all $0\leq s\leq t\leq T$ and
$x\in{\mathbb{R}}^{d}$,

$|\phi_{t,s}(x)|\leq
C(1+|x|)\quad\text{and}\quad\left\|\phi_{t,s}\right\|_{BV_{\mathrm{loc}}}\leq
C.$

Finally, if $(b^{\varepsilon})_{\varepsilon>0}$ is as in (2.3) and
$\phi^{\varepsilon}_{t,s}$ is the corresponding forward flow, then, for all
$1\leq p<\infty$,

$\lim_{\varepsilon\to
0}\phi^{\varepsilon}_{\cdot,s}=\phi_{\cdot,s}\quad\text{strongly in
}C([s,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))\quad\text{and}\quad\lim_{\varepsilon\to
0}\phi^{\varepsilon}_{t,\cdot}=\phi_{t,\cdot}\quad\text{strongly in
}C([0,t],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})),$

and the convergence also holds in the weak-$\star$ sense in
$L^{\infty}_{\mathrm{loc}}$.

###### Proof.

For $\varepsilon>0$ and $(b^{\varepsilon})_{\varepsilon>0}$ as in (2.3), it is
standard that, for $t\in(0,T]$, the vector-valued solution of

$\frac{\partial u^{\varepsilon}}{\partial s}+b^{\varepsilon}\cdot\nabla
u^{\varepsilon}=0\quad\text{in }(0,t)\times{\mathbb{R}}^{d},\quad
u^{\varepsilon}(t,x)=x$

is given by $u^{\varepsilon}(s,x)=\phi^{\varepsilon}_{t,s}(x)$ for
$s\in[0,t]$, where $\phi^{\varepsilon}$ is the flow corresponding to
$b^{\varepsilon}$. By Theorem 4.4, we have the given convergence statements,
as $\varepsilon\to 0$, of $\phi^{\varepsilon}$ to the vector valued duality
solution $u$ of (4.2) in $[0,t]\times{\mathbb{R}}^{d}$ with terminal value
$u(t,\cdot)=x$.

The flow property for smooth $b^{\varepsilon}$ yields, for $0\leq s\leq t\leq
T$ and $x\in{\mathbb{R}}^{d}$,
$\phi^{\varepsilon}_{s,t}(\phi^{\varepsilon}_{t,s}(x))$. By Lemma 2.2 and the
above strong $L^{p}$-convergence statement, we may take $\varepsilon\to 0$ to
obtain $\phi_{s,t}(u(s,x))=x$, and then, by Proposition 2.2, we must have
$u(s,x)=\phi_{t,s}(x)$. The other statements now follow immediately in view of
Theorem 4.4. Note that we are using that, for $s\in[0,T)$, the map
$[s,T]\times{\mathbb{R}}^{d}\ni(t,x)\mapsto\phi_{t,s}(x)$ is the duality
solution of the initial value problem

$\frac{\partial\tilde{u}}{\partial
t}-b(t,x)\cdot\nabla\tilde{u}=0\quad\text{in
}[s,T]\times{\mathbb{R}}^{d},\quad\tilde{u}(s,x)=x,$

whose theory can be treated exactly as for (4.2).

∎

####  4.3.2 The regular Lagrange property

We now observe that there is a representation formula for the duality solution
of the transport equation (4.2).

######  Theorem 4.7.

Let $1\leq p\leq\infty$. Then there exists a constant $C>0$ depending only on
$p$ and the constant in (2.1) such that, for all $F\in
L^{p}_{\mathrm{loc}}\cap C$, $R>0$, and $0\leq s\leq t\leq T$,

(4.8) $\left\|F\circ\phi_{t,s}\right\|_{L^{p}(B_{R})}\leq
C\left\|F\right\|_{L^{p}(B_{R+C})}.$

In particular, for any $A\subset{\mathbb{R}}^{d}$ with finite Lebesgue
measure,

(4.9) $\left|\left\\{x:\phi_{t,s}(x)\in A\right\\}\right|\leq C|A|.$

If $u_{T}\in L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})$, then the duality
solution of (4.2) is given by

(4.10) $u(t,x)=u_{T}(\phi_{T,t}(x)).$

If $u_{T}$ has a version which is continuous almost everywhere, then, for
$t<T$, $u(t,\cdot)$ also has a version that is continuous almost everywhere.

######  Remark 4.4.

When $u_{T}$ is not continuous, then (4.10) must be interpreted as the
continuous extension of the operator $u_{T}\mapsto u_{T}\circ\phi_{T,t}$ to
$u_{T}\in L^{p}_{\mathrm{loc}}$, which is well-defined in view of the estimate
(4.8).

######  Remark 4.5.

The estimate (4.9) is called the regular Lagrange property. It reinforces the
fact that $\phi_{t,s}$ does not concentrate in sets of measure zero.

######  Remark 4.6.

The propagation of almost-everywhere continuity is a consequence of the same
property for the forward flow (Proposition 2.2). Note that it is not true in
general that a function $u\in BV_{\mathrm{loc}}({\mathbb{R}}^{d})$ is
continuous almost everywhere, unless $d=1$.

###### Proof of Theorem 4.7.

For continuous $u_{T}$, the representation formula is an immediate consequence
of the renormalization property Theorem 4.5 and Theorem 4.6. The estimate
(4.8) then follows from Theorem 4.4, and (4.9) is obtained by taking $p=1$ and
$F=\mathbf{1}_{A}$.

For the claim about almost everywhere continuity, define

$A:=\left\\{y\in{\mathbb{R}}^{d}:u_{T}\text{ is not continuous at
}y\right\\}.$

Then $|A|=0$, and then (4.9) gives, for $0\leq t<T$,

$\left|\left\\{x\in{\mathbb{R}}^{d}:u_{T}\text{ is not continuous at
}\phi_{t,T}(x)\right\\}\right|=0.$

It follows that $u_{T}$ is continuous at $\phi_{t,T}(x)$ for a.e. $x$. By
Proposition 2.2, $\phi_{t,T}$ is continuous almost everywhere, and the result
follows. ∎

Recalling the duality relationship between (4.1) and (4.2) from Theorem 4.4,
we then have the following.

######  Corollary 4.2.

For any $1\leq p\leq\infty$ and $f_{0}\in
L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})$, the good solution $f$ of (4.1) is
given at time $t>0$ by $\phi_{t,0}^{\\#}f_{0}$.

######  Remark 4.7.

The regular Lagrange property says that the measure $\phi_{t,0}^{\\#}f_{0}$ is
well-defined and absolutely continuous with respect to Lebesgue measure, with
a density in $L^{p}_{\mathrm{loc}}$. If $f_{0}$ is the density for a
probability measure, that is, $f_{0}\in L^{1}_{+}({\mathbb{R}}^{d})$ and $\int
f_{0}=1$, then $f(t,\cdot)$ is the law at time $t$ of the stochastic process
$\phi_{t,0}(X)$, where $X$ is a random variable with density $f_{0}$.

A consequence of renormalization and the regular Lagrange property is the fact
that the forward flow $\phi_{t,s}$ solves the ODE (2.2) for a.e. initial
$x\in{\mathbb{R}}^{d}$. A first step is the following lemma.

######  Lemma 4.2.

For all $p\in[1,\infty)$ and $s\in[0,T)$, $\\{(t,x)\mapsto
b(t,\phi_{t,s}(x))\\}\in L^{1}([0,T],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$.
If $(b^{\varepsilon})_{\varepsilon>0}$ is as in (2.3) and
$(\phi^{\varepsilon})_{\varepsilon>0}$ is the corresponding flow, then, for
all $R>0$,

$\lim_{\varepsilon\to
0}\int_{s}^{T}\left\|b^{\varepsilon}(t,\phi^{\varepsilon}_{t,s})-b(t,\phi_{t,s})\right\|_{L^{p}(B_{R})}dt=0.$

###### Proof.

The first claim follows from (4.8): there exists $C>0$ independent of $s$ and
$R$ such that, for all $t\in[0,T]$,
$\left\|b(t,\phi_{t,s})\right\|_{L^{p}(B_{R})}\leq
C\left\|b(t,\cdot)\right\|_{L^{p}(B_{R+C})}$.

For $\delta>0$ and $0\leq s\leq t\leq T$, we write

$\displaystyle\left\|b^{\varepsilon}(t,\phi^{\varepsilon}_{t,s})-b(t,\phi_{t,s})\right\|_{L^{p}(B_{R})}$
$\displaystyle\leq\left\|b^{\varepsilon}(t,\phi^{\varepsilon}_{t,s})-b^{\delta}(t,\phi^{\varepsilon}_{t,s}\right\|_{L^{p}(B_{R})}+\left\|b^{\delta}(t,\phi^{\varepsilon}_{t,s})-b^{\delta}(t,\phi_{t,s}\right\|_{L^{p}(B_{R})}$
$\displaystyle+\left\|b^{\delta}(t,\phi_{t,s})-b(t,\phi_{t,s}\right\|_{L^{p}(B_{R})}.$

By (4.8), for some $C>0$ independent of $\delta$, $\varepsilon$, $s$, and $t$,

$\left\|b^{\varepsilon}(t,\phi^{\varepsilon}_{t,s})-b^{\delta}(t,\phi^{\varepsilon}_{t,s})\right\|_{L^{p}(B_{R})}\leq
C\left\|b^{\varepsilon}(t,\cdot)-b^{\delta}(t,\cdot)\right\|_{L^{p}(B_{R+C})}$

and

$\left\|b^{\delta}(t,\phi_{t,s})-b(t,\phi_{t,s})\right\|_{L^{p}(B_{R})}\leq
C\left\|b^{\delta}(t,\cdot)-b(t,\cdot)\right\|_{L^{p}(B_{R+C})}.$

The smoothness of $b^{\delta}$ implies that, for all $t\in[s,T]$, as
$\varepsilon\to 0$, $b^{\delta}(t,\phi^{\varepsilon}_{t,s})$ converges a.e. to
$b^{\delta}(t,\phi_{t,s})$. Sending $\varepsilon\to 0$ and using dominated
convergence, we thus have

$\limsup_{\varepsilon\to
0}\int_{s}^{T}\left\|b^{\varepsilon}(t,\phi^{\varepsilon}_{t,s})-b(t,\phi_{t,s})\right\|_{L^{p}(B_{R})}dt\leq
C\int_{s}^{T}\left\|b^{\delta}(t,\cdot)-b(t,\cdot)\right\|_{L^{p}(B_{R+C})}dt.$

The proof of the claim is finished upon sending $\delta\to 0$ and again using
dominated convergence. ∎

######  Theorem 4.8.

Fix $1\leq p<\infty$ and $s\in[0,T)$. Then

$\left\\{(t,x)\mapsto\phi_{t,s}(x)\right\\}\in
L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d},W^{1,1}([0,T])),$

and, for a.e. $x\in{\mathbb{R}}^{d}$, $[s,T]\ni t\mapsto\phi_{t,s}(x)$ is an
absolutely continuous solution of

$\phi_{t,s}(x)=x+\int_{s}^{t}b(r,\phi_{r,s}(x))dr.$

If $(b^{\varepsilon})_{\varepsilon>0}$ satisfy (2.3) and $\phi^{\varepsilon}$
is the corresponding flow, then, for all $R>0$,

$\lim_{\varepsilon\to
0}\left\|\phi^{\varepsilon}_{\cdot,s}-\phi_{\cdot,s}\right\|_{L^{p}(B_{R},W^{1,1}([s,T])}=0.$

For all $0\leq r\leq s\leq t\leq T$, $\phi_{t,r}=\phi_{t,s}\circ\phi_{s,r}$
a.e.

######  Remark 4.8.

The fact that $\partial_{t}\phi_{t,\cdot}\in L^{1}$ is due to the fact that we
are assuming the weakest possible integrability of $b$ in the time variable.
If $b\in L^{q}$ for some $q>1$, then the forward flow belongs to $W^{1,p}$ for
any $p\leq q$.

######  Remark 4.9.

The composition $\phi_{t,s}\circ\phi_{s,r}$ is made sense of due to (4.8) and
the fact that the forward flow takes values in
$L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d})$.

###### Proof of Theorem 4.8.

For $\varepsilon>0$, we have
$\partial_{t}\phi^{\varepsilon}_{t,s}(x)=b^{\varepsilon}(t,\phi^{\varepsilon}_{t,s}(x))$.
By Lemma 4.2, sending $\varepsilon\to 0$, we see that the distribution
$\partial_{t}\phi_{t,s}(x)$ satisfies, in the distributional sense,
$\partial_{t}\phi_{t,s}(x)=b(t,\phi_{t,s}(x))$, and therefore, for all $R>0$,

$\left\|\int_{s}^{T}|\phi_{t,s}|dt\right\|_{L^{p}(B_{R})}\leq\int_{s}^{T}\left\|\phi_{t,s}\right\|_{L^{p}(B_{R})}dt<\infty.$

The convergence claim and the solvability of the ODE follow immediately in
view of the fact that $\phi^{\varepsilon}_{s,s}(x)=\phi_{s,s}(x)=x$ for all
$\varepsilon>0$ and $x\in{\mathbb{R}}^{d}$.

To prove the last claim, we note that the equality
$\phi_{r,t}\circ\phi_{t,r}=\operatorname{Id}$ holds as functions in
$L^{p}_{\mathrm{loc}}$, and, in view of the flow property of the backward
flow,

$\phi_{r,t}\circ(\phi_{t,s}\circ\phi_{s,r})=\phi_{r,s}\circ\phi_{s,t}\circ\phi_{t,s}\circ\phi_{s,r}=\phi_{r,s}\circ\phi_{s,r}=\operatorname{Id}.$

It follows from Proposition 2.2 that $\phi_{t,r}=\phi_{t,s}\circ\phi_{s,r}$
a.e., as desired. ∎

We recall that Proposition 2.2 implies that any right-inverse of the backward
flow is determined uniquely almost everywhere. We remark here that this
property actually follows from the duality between the transport and
continuity equations.

######  Theorem 4.9.

Assume $\psi\in C([0,t],L^{p}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ satisfies
$\phi_{s,t}(\psi_{s}(x))=x$ for all $s\in[0,t]$, for a.e.
$x\in{\mathbb{R}}^{d}$. Then $\psi=\phi_{t,\cdot}$.

###### Proof.

It suffices to show that $u(t,x)=\psi_{s}(x)$ is the unique (vector-valued)
duality solution of (4.2) with terminal data equal to $x$ at time $t$.

Fix $g\in C_{c}({\mathbb{R}}^{d})$. For a.e. $x\in{\mathbb{R}}^{d}$, if
$y=\psi_{t}(x)$, we have $\phi_{s,t}(y)=x$ by assumption. Therefore, the
change of variables formula yields

$\int_{{\mathbb{R}}^{d}}g(x)\psi_{t}(x)dx=\int_{{\mathbb{R}}^{d}}g(\phi_{s,t}(y))yJ_{s,t}(y)dy=\int_{{\mathbb{R}}^{d}}f(t,y)ydy,$

where $f$ is the good solution of the forward continuity equation with initial
condition $g$ at time $s$. ∎

######  Remark 4.10.

A corresponding result characterizing $\phi_{\cdot,s}$ on $[s,T]$ follows in
exactly the same way, by considering the duality between the IVP and TVP for,
respectively, an appropriate transport and continuity equation.

######  Remark 4.11.

The uniqueness result above demonstrates that the right-inverse property is a
crucial property of the forward flow. In other words, it implies that
$\phi_{t,s}$ solves the ODE, that it solves the transport PDE in the duality
sense, and that it has the regularity properties laid out in Theorems 4.7 and
4.8.

###  4.4 Characterizations

We now present alternative ways to characterize the solutions of the forward
continuity and backward transport equations identified above. Although the PDE
(4.2) does not make sense as a distribution, we nevertheless can characterize
solutions in a PDE sense through the use of $\sup$\- and $\inf$-convolutions.
The propagation of almost-everywhere continuity proved in Theorem 4.7 is a
crucial ingredient.

By using this characterization in duality with the conservative equation, we
then show that nonnegative distributional solutions of (4.1) are unique, and
therefore equal to the “good” solution identified by the formula (4.3). As a
consequence, we finally conclude with the uniqueness of regular Lagrangian
flows, forward in time, of the ODE (2.2).

####  4.4.1 The nonconservative equation: $\sup$ and $\inf$ convolutions

Given $\delta>0$ and $u\in L^{\infty}({\mathbb{R}}^{d})$, we define the
$\sup$\- and $\inf$-convolutions

$u^{\delta}(x):=\operatorname*{ess\,sup}_{y\in{\mathbb{R}}^{d}}\left\\{u(y)-\frac{1}{2\delta}|x-y|^{2}\right\\}$

and

$u_{\delta}(x):=\operatorname*{ess\,inf}_{y\in{\mathbb{R}}^{d}}\left\\{u(y)+\frac{1}{2\delta}|x-y|^{2}\right\\}.$

These regularizations are common in the theory of viscosity solutions, or
generally for equations satisfying a maximum principle in spaces of continuous
functions. The supremum and infimum must be essential, because $u$ is only
defined almost everywhere.

######  Lemma 4.3.

Assume that $u\in L^{\infty}({\mathbb{R}}^{d})$ is continuous almost
everywhere. Then, for all $\delta>0$, $u_{\delta},u^{\delta}$ are globally
Lipschitz with constant

$(\operatorname*{ess\,sup}u-\operatorname*{ess\,inf}u)^{1/2}\delta^{-1/2},$

and

$u_{\delta}\leq u\leq u^{\delta}\quad\text{a.e.}$

As $\delta\to 0$, $u^{\delta}$ decreases to $u$ and $u_{\delta}$ increases to
$u$ a.e. Finally, the $\operatorname*{ess\,sup}$ and
$\operatorname*{ess\,inf}$ in the definitions of $u^{\delta}$ and $u_{\delta}$
can be restricted to respectively $y\in B_{R^{\delta}(x)}(x)$ and
$B_{R_{\delta}(x)}(x)$, where

$R^{\delta}(x)=2(u^{2\delta}(x)-u^{\delta}(x))^{1/2}\delta^{1/2}.$

and

$R_{\delta}(x)=2(u_{\delta}(x)-u_{2\delta}(x))^{1/2}\delta^{1/2}.$

###### Proof.

Fix $x\in{\mathbb{R}}^{d}$ and $r>0$. We thus have

$u^{\delta}(x)\geq\operatorname*{ess\,sup}_{y\in
B_{r}(x)}u(y)-\frac{r^{2}}{2\delta}.$

Sending $r\to 0$, we see that $u^{\delta}(x)\geq u(t,x)$ whenever $u$ is
continuous at $x$, and therefore $u^{\delta}\geq u$ a.e. Similarly,
$u_{\delta}\leq u$ a.e.

We now observe that, if
$R>(\operatorname*{ess\,sup}u-\operatorname*{ess\,inf}u)^{1/2}$, then, for
a.e. $y\notin B_{R\delta^{1/2}}$,

$u(y)-\frac{|x-y|^{2}}{2\delta}\leq\operatorname*{ess\,sup}u-R^{2}<\operatorname*{ess\,inf}u\leq
u^{\delta}(x).$

By also using a similar argument for $u_{\delta}$, we see that

$u^{\delta}(x):=\operatorname*{ess\,sup}_{|y-x|\leq
R\delta^{1/2}}\left\\{u(y)-\frac{1}{2\delta}|x-y|^{2}\right\\}$

and

$u_{\delta}(x):=\operatorname*{ess\,inf}_{|y-x|\leq
R\delta^{1/2}}\left\\{u(y)+\frac{1}{2\delta}|x-y|^{2}\right\\}.$

It is then straightforward to see that $u^{\delta}$ and $u_{\delta}$ are
respectively decreasing and increasing pointwise as $\delta$ decreases to $0$,
and converge whenever $u$ is continuous at $x$ (and thus a.e.) to $u(x)$.

For fixed $x\in{\mathbb{R}}^{d}$, $\delta>0$, and $\eta>0$, define

$A_{\delta,\eta}(x):=\left\\{y\in{\mathbb{R}}^{d}:u(y)-\frac{|x-y|^{2}}{2\delta}>u^{\delta}(x)-\eta\right\\}.$

Then, by definition, $A_{\delta,\eta}(x)$ has nonzero Lebesgue measure.
Therefore, for any $x^{\prime}\in{\mathbb{R}}^{d}$, there exists $y\in
A_{\delta,\eta}(x)$ such that

$u(y)-\frac{|x^{\prime}-y|^{2}}{2\delta}\leq u^{\delta}(x^{\prime}),$

and so

$u^{\delta}(x)-u^{\delta}(x^{\prime})\leq\frac{|x^{\prime}-y|^{2}}{2\delta}-\frac{|x-y|^{2}}{2\delta}+\eta\leq\frac{R}{\delta^{1/2}}|x^{\prime}-x|+\frac{|x^{\prime}-x|^{2}}{\delta}+\eta.$

Switching the roles of $x$ and $x^{\prime}$ and using the fact that $\eta$ was
arbitrary, we see that, for all $x\in{\mathbb{R}}^{d}$,

$\limsup_{x^{\prime}\to
x}\frac{|u^{\delta}(x^{\prime})-u^{\delta}(x)|}{|x^{\prime}-x|}\leq\frac{R}{\delta^{1/2}}.$

We may then let $R$ decrease down to
$(\operatorname*{ess\,sup}u-\operatorname*{ess\,inf}u)^{1/2}$, and the same
proof for $u_{\delta}$ holds.

For any $\eta>0$ and a.e. $y\in A^{\eta}_{\delta}$,

$u^{2\delta}(x)\geq
u(y)-\frac{|x-y|^{2}}{4\delta}>u^{\delta}(x)+\frac{|x-y|^{2}}{2\delta}-\eta,$

and so

$|y-x|\leq 2(u^{2\delta}(x)-u^{\delta}(x)+\eta)^{1/2}\delta^{1/2}.$

Therefore, for a.e. $y$ such that $|y-x|>R^{\delta}(x)$, we must have
$u(y)-\frac{|x-y|^{2}}{2\delta}<u^{\delta}(x)$, and the statement about
restricting the $\operatorname*{ess\,sup}$ follows. The corresponding result
for $u_{\delta}$ is proved in the same way. ∎

######  Theorem 4.10.