# Entropy-minimizing dynamical transport on Riemannian manifolds

Gabriele Bocchi Department of Mathematics, University of Rome Tor Vergata. Via
della Ricerca Scientifica 1, 00133 Roma, Italy<EMAIL_ADDRESS>Alessio
Porretta Department of Mathematics, University of Rome Tor Vergata. Via della
Ricerca Scientifica 1, 00133 Roma, Italy<EMAIL_ADDRESS>
###### Abstract

Given a smooth Riemannian manifold $(M,g)$, compact and without boundary, we
analyze the dynamical optimal mass transport problem where the cost is given
by the sum of the kinetic energy and the relative entropy with respect to a
reference volume measure $e^{-V}dx$. Under the only assumption that the
prescribed marginals lie in $L^{1}(M)$, and a lower bound on the Ricci
curvature, we characterize the minimal curves as unique weak solutions of the
optimality system coupling the continuity equation with a backward Hamilton-
Jacobi equation (with source given by $\log(m)$). We give evidence that the
entropic cost enhances diffusive effects in the evolution of the optimal
densities, proving $L^{1}\to L^{\infty}$ regularization in time for any
initial-terminal data, and smoothness of the solutions whenever the marginals
are positive and smooth. We use displacement convexity arguments (in the
Eulerian approach) and gradient bounds from quasilinear elliptic equations. We
also prove the convergence of optimal curves towards the classical Wasserstein
geodesics, as the entropic term is multiplied by a vanishing parameter,
showing that this kind of functionals can be used to build a smoothing
approximation of the standard optimal transport problem.

##  1 Introduction

Let $(M,g)$ be a smooth, connected $d$-dimensional Riemannian manifold,
assumed to be compact and without boundary, endowed with a metric tensor
$g=(g_{ij})$ and a volume form $dx$. We denote by ${\mathcal{P}}(M)$ the set
of probability measures on $M$. Given a time horizon $T>0$ and two fixed
(initial and terminal) measures $m_{0},m_{1}\in{\mathcal{P}}(M)$, we analyze
in this note the optimal transport problem

$\displaystyle\min\,\,\mathcal{F}_{\varepsilon}(m,v)\coloneqq\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}\left|v\right|^{2}\,dm+\varepsilon\int_{0}^{T}{\mathcal{H}}(m(t);\nu)\,,\qquad$
(1.1) $\displaystyle\qquad\hbox{among all
}\quad(m,v)\quad\hbox{}:\begin{cases}\partial_{t}m-div_{g}(vm)=0\\\
m(0)=m_{0}\,,\,\,m(T)=m_{1}\end{cases}$

where $\nu:=e^{-V(x)}dx$ and

${\mathcal{H}}(m;\nu)=\int_{M}\log\left(\frac{dm}{d\nu}\right)dm=\int_{0}^{T}\\!\\!\\!\\!\int_{M}m(\log
m+V)\,dxdt$

denotes the relative entropy of $m$ with respect to a reference measure
$e^{-V}dx$, given for some Lipschitz continuous vector field $V$.

In (1.1), $m(t)$ is an (absolutely continuous) arc joining $m_{0}$ and $m_{1}$
with velocity $v$, $\left|\cdot\right|$ is the length of vector fields and
$div_{g}(\cdot)$ the intrinsic divergence operator on the manifold $M$. The
functional (1.1) can be seen as a perturbation of the kinetic energy
functional used in the dynamical version of mass transportation [2]. The
additional term in (1.1) prevents concentration effects by penalizing the
relative entropy and is supposed to enhance some form of dissipation along
optimal curves. This is only one, yet very natural, among possibly different
entropic regularizations of the classical optimal transport energy. In this
respect, it follows a stream of research which has been very intensive in
recent times, where other kind of regularizations of the Wasserstein distance
were suggested (see [13], [17], [30], [34]).

The evolution of optimal transport densities with additional costs that
consider the effects of congestion has been exploited so far in several
directions, see e.g. [3], [5], and especially [27], where some
$L^{1}-L^{\infty}$ regularization in the time evolution of the optimal curves
was proved using variational techniques. Similar problems were addressed in
[7], [8], [9], [10], [20], [41] with a different approach based on ideas
coming from mean field game theory and PDE estimates on the optimality system
(state-adjoint state) associated to (1.1), which is

$\begin{cases}-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}=\varepsilon(\log(m)+V(x))\quad&\text{in $(0,T)\times M$}\\\
\partial_{t}m-div_{g}(m\nabla u)=0&\text{in $(0,T)\times M$}\\\
m(0)=m_{0},\quad m(T)=m_{1}&\text{in $M$.}\end{cases}$ (1.2)

As firstly observed by P.-L. Lions [35], (1.2) is just one instance of PDE
system appearing in mean field game theory ([25], [26]) and some smoothness on
the optimal curves of this kind of functionals can be derived from gradient
estimates on the adjoint state $u$ (the so-called Kantorovich potential, in
mass transportation language). This approach, relying on the ellipticity
hidden in the optimaliy system, was thoroughly developed in [39], [40], [43].
In particular, the case of functional (1.1), with the additional entropy term,
was addressed in [43] for convex domains of $\mathbb{R}^{d}$ (with no-flux
condition at the boundary) assuming that the marginals $m_{0},m_{1}$ are
positive and smooth, in which case the minima can be proved to be positive and
smooth for all times. Similar results were also proved for Gaussian-like
measures in the whole Euclidean space.

The goal of this paper is twofold. First of all, we give some general result
on problem (1.1), under the only assumption that the marginals $m_{0},m_{1}\in
L^{1}(M)$. In particular, the marginals do not need to be positive (nor
smooth), extending many results of [43] to the case of nonnegative initial-
terminal data. Except for some special results obtained in one dimension [11],
the case of compactly supported marginals had not been developed, so far.

Secondly, we analyze the problem in the setting of a Riemannian manifold, in
order to get a more exhaustive comprehension of some crucial tools. In fact,
in the genuine optimal transport viewpoint, it is well understood ([14], [15],
[16], [36], [38], [47], [49]) that the Riemannian setting is the most natural
to observe the role of Ricci curvature in regularity arguments related to
displacement convexity and entropy dissipation. In our results, we will
require the only condition that the Ricci curvature is bounded below, and we
will obtain estimates which are totally consistent with the pure mass
transport problem, embedding therefore the case $\varepsilon=0$ in (1.1) into
a family of similar problems. As an example, if $Ric(M)\geq\Lambda I$, we will
prove the diplacement $\Lambda-$ convexity of the entropy along Wasserstein
geodesics as the limit of $\Lambda_{\varepsilon}$-convexity along the optimal
curves of (1.1).

Concerning the characterization of minimal curves of (1.1), we summarize our
main results in the following statement.

######  Theorem 1.1:

Let $(M,g)$ be a smooth compact Riemannian manifold without boundary, with
$Ric_{g}(M)$ bounded below. Let $m_{0},m_{1}\in L^{1}(M)\cap\mathcal{P}(M)$,
and assume that $V\in W^{2,\infty}(M)$, $\varepsilon>0$. Then the functional
$\mathcal{F}_{\varepsilon}$ in (1.1) admits a unique minimum, given by
$(m,\nabla u)$, where $(m,u)$ is the unique weak solution (in the sense of
Definition 5.1) of the system (1.2), with $\int_{M}u(T)m_{1}=0$. Moreover we
have:

  * (i)

$m>0$ a.e. in $(0,T)\times M$.

  * (ii)

$u,m\in L^{\infty}_{loc}((0,T)\times M)$ and $u(0)\in L^{1}(dm_{0}),u(T)\in
L^{1}(dm_{1})$.

  * (iii)

if $m_{0},m_{1}\in W^{1,\infty}(M)$ and are (strictly) positive, and if $V\in
C^{k,\alpha}(M)$, then $u\in C^{k+1,\alpha}((0,T)\times M),m\in
C^{k,\alpha}((0,T)\times M)$.

The statement of Theorem 1.1 summarizes several different results that we
establish later. In fact, we will start from the case of smooth and positive
marginals (see Theorem 5.4), proving the existence of smooth solutions to the
system (1.2). To this purpose, we follow the strategy suggested by P.-L. Lions
[35], and developed in [40], [43], which consists in rewriting the system
(1.2) as a quasilinear elliptic equation for $u$ (see (3.16)) and using the
continuity method, relying on gradient bounds, in order to produce a smooth
solution $u$. Following [43], this strategy is first employed for a penalized
auxiliary problem (4.14), where the $L^{\infty}-$ norm of $u$ is readily
controlled. Then a compactness argument yields a smooth solution $(m,u)$ after
a suitable normalization of $u$.

Once we have built smooth solutions of (1.2), we will obtain all relevant
estimates which remain robust for merely $L^{1}$ marginals $m_{0},m_{1}$. This
step includes a bound on the minimal value of $\mathcal{F}_{\varepsilon}$,
obtained by building suitable competitors, which in turn will yield local
uniform bounds on $u$.

From the local bounds on $u$ we will also obtain the local $L^{\infty}$\-
bound on $m$. Notice that this $L^{1}-L^{\infty}$ regularization on the
density is not a straightforward extension from the euclidean case, because
the Ricci curvature is allowed to be negative. In particular, the $L^{\infty}$
control on $m$ does not follow directly from the displacement convexity
inequalities as in [43].

Finally, we will derive local (in time) bounds for $Du$ in $L^{2}$ (from the
HJ equation) and for the Fisher information of $m$ (by displacement convexity
estimates) that yield the suitable compactness arguments. This latter step
includes a relaxation result on the system and the convergence to weak
solutions, providing with the final characterization of the minima of (1.1)
(see Theorem 5.10).

Many of those tools, involving estimates and stability on the optimality
system (1.2), are stable as $\varepsilon\to 0$.

Indeed, we conclude the article by giving a result of convergence of the
optimal curves towards the Wasserstein geodesic, as $\varepsilon\to 0$, as
well as the convergence of $\min\mathcal{F}_{\varepsilon}$ towards
$\min\mathcal{F}_{0}$. This convergence occurs with a rate $O(\varepsilon)$
when the marginals have finite entropy, while we cannot prove a rate better
than $O(\varepsilon|\log\varepsilon|)$ for the general case of marginals only
in $L^{1}(M)$.

######  Theorem 1.2:

Under the assumptions of Theorem 1.1, let $(m_{\varepsilon},\nabla
u_{\varepsilon})$ be the minima of (1.1), where
$(m_{\varepsilon},u_{\varepsilon})$ solves (1.2), with
$\int_{M}u_{\varepsilon}(T)m_{1}=0$, and let $(m,\nabla u)$ be the Wasserstein
geodesic between $m_{0},m_{1}$. Then, as $\varepsilon\to 0$, we have

$\displaystyle m_{\varepsilon}$ $\displaystyle\to m\quad\hbox{in
$C^{0}([0,T],{\mathcal{P}}(M))$ and weakly in $L^{1}((0,T)\times M)$,}$
$\displaystyle m_{\varepsilon}\nabla u_{\varepsilon}$ $\displaystyle\to
m\nabla u\quad\hbox{weakly in $L^{1}((0,T)\times M)$,}$

and $\min\mathcal{F}_{\varepsilon}\to\min\mathcal{F}_{0}$. In particular we
have

$\min\mathcal{F}_{\varepsilon}=\min\mathcal{F}_{0}+r_{\varepsilon}$

where $r_{\varepsilon}=O(\varepsilon)$ if $m_{0},m_{1}$ have finite entropy,
otherwise $r_{\varepsilon}=O(\varepsilon|\log\varepsilon|)$.

Last but not least, we will show (see Theorem 6.4) that using a suitable
approximation of $m_{0},m_{1}$ with sequences
$m_{0\varepsilon},m_{1\varepsilon}$ of smooth positive functions, the
Wasserstein geodesic between $m_{0},m_{1}$ can be approximated by smooth
minimizers of $\mathcal{F}_{\varepsilon}$, in a way that $u_{\varepsilon}$
remains uniformly bounded in Lipschitz norm. Hence, in particular, the
Kantorovich potentials converge uniformly in $M$.

This latter result shows that the purpose of smoothing the Wasserstein
geodesic can be fully accomplished with the functional (1.1); in particular,
this gives a general Eulerian strategy towards the proof of displacement
convexity properties of the geodesics of optimal transport. As an example, we
recover some results of [14], [15] with an alternative proof, avoiding the use
of EVI inequalities in favor of a standard Eulerian approach which is now
justified going through problems (1.1).

##  2 Notations and setting of the problem

In the following, we recall some elements of Riemannian geometry (see e.g.
[46]). Throughout the paper, $(M,g)$ denotes a smooth, compact, connected and
oriented $d$-dimensional Riemannian manifold without boundary, with metric
tensor $g=(g_{ij})$, inverse $g^{-1}=(g^{ij})$ and determinant
$\left|g\right|$. The orientation induces a unitary volume form $dx$. If $w$
and $v$ are two vector fields on $M$, we denote by

$w\operatorname{\cdot_{g}\\!}w\coloneqq\sum_{ij}g_{ij}(x)w_{i}v_{j}$

their scalar product in the tangent space $T_{x}\\!M$. The length of a vector
field is given by $|w|=\sqrt{w\operatorname{\cdot_{g}\\!}w}$. Correspondingly,
there is a scalar product in the cotangent space $T^{*}_{x}\\!M$, which is
defined on differential $1$-forms $\omega$ and $\nu$ on $M$ as
$\omega\operatorname{\cdot_{g}\\!}\nu\coloneqq\sum_{ij}g^{ij}(x)\omega_{i}\nu_{j}$.

Let $x_{j}$, $j=1,\dots,d$, be a local system of coordinates: if $u\in
C^{1}(M)$, the covariant gradient of $u$, denoted by $\nabla u$, is the vector
field with coordinates $\nabla_{i}u=g^{ij}(x)u_{x_{j}}$. Therefore, given
$u,v\in C^{1}(M)$,

$\nabla u\operatorname{\cdot_{g}\\!}\nabla
v=\sum_{ij}g^{ij}(x)u_{x_{i}}v_{x_{j}}$

We denote the Levi-Civita connection associated to the metric $g$ with the
letter $D$ and we will derivate covariantly vector and tensor fields on $M$.
Recalling that, in local coordinates, the Christoffel symbols are

$\Gamma^{k}_{ij}=\frac{1}{2}\sum_{l}\left(\frac{\partial g_{jl}}{\partial
x_{i}}+\frac{\partial g_{li}}{\partial x_{j}}-\frac{\partial g_{ij}}{\partial
x_{l}}\right)g^{lk}\,,$

the covariant derivative of a $C^{1}$ vector field $X=(X_{j})$ along the
vector field $v=(v_{i})$ is the vector field $D_{v}X$ with $k$-th coordinate
given by

$(D_{v}X)_{k}=\sum_{ij}v_{i}X_{j}\Gamma^{k}_{ij}+(\nabla
X_{k})\operatorname{\cdot_{g}\\!}v.$

If $X=(X_{j})$ is a $C^{1}$ vector field on $M$, the divergence of $X$ is
defined by

$div_{g}X=\frac{1}{\sqrt{\left|g\right|}}\sum_{k}(\sqrt{\left|g\right|}X^{k})_{x_{k}}$

and the Leibniz rule: $div_{g}(fX)=\nabla
f\operatorname{\cdot_{g}\\!}X+fdiv_{g}X$ holds for every $f\in C^{1}(M)$ and
any $C^{1}$ vector field $X$ on $M$. Furthermore, by the Stokes theorem, we
have

$\int_{M}div_{g}X\,dx=0\,.$

The Hessian $\nabla^{2}u$ of a $C^{2}$ function $u$ is the symmetric
$2$-tensor given by

$(\nabla^{2}u)(v,w)\coloneqq(D_{v}\nabla
u)\operatorname{\cdot_{g}\\!}w=(D_{w}\nabla u)\operatorname{\cdot_{g}\\!}v$

for every vector fields $v,w$ on $M$, where the last equality follows by the
symmetry and the compatibility with the metric of the Levi-Civita connection.
The components of the Hessian are the second covariant derivatives, given by

$\nabla_{ij}u=u_{x_{i}x_{j}}-\sum_{k}\Gamma^{k}_{ij}u_{x_{k}}.$

In particular for every $C^{2}$ functions $f,u$ and every vector field $v$ it
holds

$v\operatorname{\cdot_{g}\\!}\nabla(\nabla f\operatorname{\cdot_{g}\\!}\nabla
u)=(\nabla^{2}f)(v,\nabla u)+(\nabla^{2}u)(v,\nabla f).$

As usual, we denote $\Delta_{g}u=div_{g}(\nabla u)$ the Laplace-Beltrami
operator on $M$. We recall the Böchner formula (see e.g. [4]):

$\tfrac{1}{2}\Delta_{g}\left|\nabla
f\right|^{2}=\left|\nabla^{2}f\right|^{2}+\nabla(\Delta_{g}f)\operatorname{\cdot_{g}\\!}\nabla
f+Ricc_{g}(\nabla f,\nabla f)$ (2.1)

for every $f\in C^{3}(M)$, where $Ricc_{g}$ is the curvature tensor of the
metric $g$. Throughout all the paper, we will assume that $Ric_{g}(M)$ is
bounded below, i.e.

$Ricc_{g}(X,X)\geq-\lambda\left|X\right|^{2}$ (2.2)

for every vector field $X$ on $M$, for some $\lambda\geq 0$. Given $M$, we
denote by ${\mathcal{P}}(M)$ the space of probability measures on $M$, endowed
with the Wasserstein metric. The characterization of the Wasserstein geodesics
in terms of optimal mass transportation on $M$ is well established, since $M$
is compact, see [38], [49]. We will always identify the measures with their
densities when they are absolutely continuous with respect to the unitary
volume measure $dx$. With $L^{p}(M)$ we indicate the standard Lebesgue space,
for $p\in[1,+\infty]$, and with $W^{k,p}(M)$ the Sobolev space of functions
with $k$ weak $L^{p}$-derivatives. We will use the Sobolev and Poincaré-
Wirtinger inequalities on $M$, for which we refer to [22]. Finally, throughout
the paper we denote $Q_{T}$ the cylinder $(0,T)\times M$, and
$\overline{Q}_{T}=[0,T]\times M$.

###  2.1 Optimal transport functional

We now make precise the sense of the minimization problem (1.1).

######  Definition 2.1:

Let $m_{0},m_{1}\in{\mathcal{P}}(M)$. A couple $(m,v)$ is a solution of the
continuity equation

$\begin{cases}\partial_{t}m-div_{g}(vm)=0\\\
m(0)=m_{0}\,,\,\,m(T)=m_{1}\,,\end{cases}$ (2.3)

if $m\in C([0,T];{\mathcal{P}}(M))$ with $m(0)=m_{0}$ and $m(T)=m_{1}$,
$v(t,x)$ is a measurable vector field on $Q_{T}$ such that
$\int_{0}^{T}\\!\\!\\!\\!\int_{M}|v|^{2}\,dm<\infty$ and the following
equality holds

$\int_{M}\varphi(t)dm(t)-\int_{M}\varphi(s)dm(s)+\int_{s}^{t}\\!\\!\\!\\!\int_{M}\left(-\partial_{t}\varphi+v\operatorname{\cdot_{g}\\!}\nabla\varphi\right)\,dm=0\,,$

for every $0\leq s<t\leq T$ and every function $\varphi\in
C^{1}(\overline{Q}_{T})$.

We recall (see [1]) that weak solutions as defined above are essentially
equivalent to absolutely continuous curves from $[0,T]$ into
${\mathcal{P}}(M)$ which have $L^{2}$ metric derivative. We also recall that
any convex, superlinear function $F(r)$ induces a lower semicontinuous
functional on the space of probability measures:

$F(m):=\begin{cases}\int_{M}F(m)\,dx&\hbox{if $m$ is absolutely continuous}\\\
+\infty&\hbox{otherwise.}\end{cases}$

Similar kind of functionals have been extensively studied, see e.g. [27] and
references therein. Even if we could consider general functions $F$, for the
sake of clarity we restrict the analysis in this paper to the specific
entropic case, in which $F(m)=m\log(m)$, and more generally to the relative
entropy in terms of a possibly inhomogeneous reference measure
$\nu=e^{-V(x)}dx$:

${\mathcal{H}}(m;\nu):=\int_{M}F\left(\frac{dm}{d\nu}\right)d\nu=\int_{M}\log\left(\frac{dm}{d\nu}\right)dm=\int_{M}m(\log
m+V)dx\,$ (2.4)

with the convention that ${\mathcal{H}}(m;\nu)=+\infty$ whenever $m$ is not
absolutely continuous with respect to $dx$. In what follows, we assume that
$V$ is (at least) Lipschitz continuous on $M$. Thanks to Definition 2.1, the
meaning of the optimal transport problem (1.1) is now clarified, to be read as

$\displaystyle\min\mathcal{F}_{\varepsilon}(m,v)\coloneqq\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}\left|v\right|^{2}\,\,dm+\varepsilon\int_{0}^{T}{\mathcal{H}}(m;\nu)\,,\qquad\nu:=e^{-V(x)}dx$
(2.5) $\displaystyle\qquad\hbox{among all
}\quad(m,v)\,:\quad\begin{cases}\partial_{t}m-div_{g}(vm)=0\\\
m(0)=m_{0}\,,\,\,m(T)=m_{1}\end{cases},$

where the equation is understood as above.

We first establish that, for every $m_{0},m_{1}\in{\mathcal{P}}(M)$, there
exists an arc along which the above functional is finite, so it admits a
finite minimum. In addition, we can give a universal upper bound on the
minimal value of $\mathcal{F}_{\varepsilon}$.

######  Proposition 2.2:

Let (2.2) hold true. There exists a constant $C(M,d,\lambda,T,\|V\|_{\infty})$
(depending on $M$ as well as on $d,\lambda,T,\|V\|_{\infty}$) such that

$\min\mathcal{F}_{\varepsilon}\leq C(M,d,\lambda,T,\|V\|_{\infty})$ (2.6)

for every $m_{0},m_{1}\in{\mathcal{P}}(M)$, and every $\varepsilon\leq 1$.

###### Proof.

Consider the heat kernel $p_{t}(x,y)$ associated to the volume measure $dx$
and the curve $\mu_{0}(\cdot):[0,1]\to{\mathcal{P}}(M)\cap C^{\infty}_{+}(M)$
generated by the heat semigroup $S_{t}$:

$t\to\mu_{0}(t,x)=S_{t}(m_{0})\coloneqq\int_{M}p_{t}(x,y)\,dm_{0}(y).$

It is a classical result (cfr. [21, Chapters 7, 8]) that $\mu_{0}(\cdot)$ is
well defined and is a smooth solution of the heat equation

$\frac{\partial}{\partial t}\mu_{0}=\Delta_{g}\mu_{0}\qquad\hbox{on
$[0,\infty)\times M$.}$

In particular we have $\mu_{0}(t,\cdot)>0$ for every $t>0$ by the strong
maximum principle. It follows that the velocity of such curve for $t>0$ is
given by the vector field

$\nu_{0}(t,x)\coloneqq\frac{\nabla\mu_{0}}{\mu_{0}}\,.$

By the Li-Yau inequality [28, Theorem 1.4] we know that there exists a
constant $C(d,\lambda)$ such that

$\frac{\left|\nabla\mu_{0}\right|^{2}}{\mu_{0}}-2\frac{\partial}{\partial
t}\mu_{0}\leq(C+\frac{2d}{t})\,\mu_{0}\,.$

Recalling that $\mu_{0}$ is a probability density for every $t>0$, integrating
the above inequality we get

$\int_{M}\left|\nu_{0}\right|^{2}\mu_{0}\,dx\leq C+\frac{2d}{t}.$ (2.7)

We now study the decay of the entropy along the heat flow. We recall that

$\frac{\partial}{\partial
t}\int_{M}\mu_{0}\log(\mu_{0})\,dx=\frac{\partial}{\partial
t}\int_{M}(\mu_{0}\log(\mu_{0})-\mu_{0})\,dx=-\int_{M}\frac{\left|\nabla\mu_{0}\right|^{2}}{\mu_{0}}\,dx\,.$

By Sobolev and Poincaré-Wirtinger inequality we have (for
$2^{*}=\frac{2d}{d-2}$ if $d>2$, or $2^{*}$ any sufficient large number if
$d=2$)

$\displaystyle\int_{M}\frac{\left|\nabla\mu_{0}\right|^{2}}{\mu_{0}}\,dx$
$\displaystyle=4\int_{M}\left|\nabla\sqrt{\mu_{0}}\right|^{2}\,dx\geq
C_{S}\left(\int_{M}\left|\sqrt{\mu_{0}}-Vol(M)^{-1}\\!\\!\int_{M}\sqrt{\mu_{0}}\,dx\right|^{2^{*}}dx\right)^{\frac{2}{2^{*}}}$
$\displaystyle\geq
c_{1}(\int_{M}\sqrt{\mu_{0}}^{2^{*}}dx)^{\frac{2}{2^{*}}}-c_{2}(\int_{M}\sqrt{\mu_{0}}\,dx)^{2}$
$\displaystyle\geq
c_{1}(\int_{M}\sqrt{\mu_{0}}^{2^{*}}dx)^{\frac{2}{2^{*}}}-c_{2}Vol(M)\,.$

By the concavity of the $\log$ function and Jensen inequality for the
probability measure $\mu_{0}$

$\begin{split}\log\left(\int_{M}\frac{1}{\mu_{0}}\left|\nabla\mu_{0}\right|^{2}\,dx+c_{2}Vol(M)\right)&\geq\frac{2}{2^{*}}\log\left(\int_{M}\sqrt{\mu_{0}}^{2^{*}}dx\right)+\log(c_{1})\\\
&\geq\frac{2}{2^{*}}\int_{M}\log\left(\mu_{0}^{\frac{2^{*}-2}{2}}\right)\mu_{0}dx+\log(c_{1})\\\
&=\frac{2}{d}\int_{M}\log\left(\mu_{0}\right)\mu_{0}dx+\log(c_{1})\,.\end{split}$
(2.8)

In other words, if $\varphi(t)\coloneqq\int_{M}\mu_{0}\log\mu_{0}\,dx$, then
we deduce

$\varphi^{\prime}(t)\leq-c_{1}e^{\frac{2}{d}\varphi}+C$

for a constant $C$ depending only on $Vol(M)$ and $d$. This implies

$(\varphi^{\prime}(t)-C)e^{-\frac{2}{d}(\varphi-Ct)}\leq-
c_{1}e^{\frac{2C}{d}t}\leq-c_{1}$

and then, integrating in $(t_{0},t_{1})$, we get

$-\frac{d}{2}e^{-\frac{2}{d}(\varphi(t_{1})-Ct_{1})}+\frac{d}{2}e^{-\frac{2}{d}(\varphi(t_{0})-Ct_{0})}+c_{1}(t_{1}-t_{0})\leq
0\,.$

In particular, letting $t_{0}\to 0$ we deduce

$-\frac{d}{2}e^{-\frac{2}{d}(\varphi(t_{1})-Ct_{1})}+c_{1}t_{1}\leq 0.$

Since $t_{1}$ is arbitrary, this means that

$\int_{M}\mu_{0}(t)\log\mu_{0}(t)\,dx=\varphi(t)\leq-\frac{d}{2}\log\left(\frac{2c_{1}}{d}t\right)+C(d,M)t$
(2.9)

for every $t>0$.
Now, for any given $\beta>1$, we consider the reparametrization
$\tilde{\mu}_{0}(t,\cdot)\coloneqq\mu_{0}(t^{\beta})$ for every $t>0$. Its
velocity field is

$\tilde{\nu}_{0}(t,\cdot)\coloneqq\beta t^{\beta-1}\nu_{0}(t^{\beta},\cdot)$

so, for any fixed $0<\delta_{0}<\frac{T}{3}$, by (2.7)

$\displaystyle\int_{0}^{\delta_{0}}\\!\\!\int_{M}\left|\tilde{\nu}_{0}\right|^{2}\tilde{\mu}_{0}\,dxdt$
$\displaystyle=\frac{1}{\beta}\int_{0}^{\delta_{0}^{\beta}}\\!\\!t^{1-\frac{1}{\beta}}\int_{M}\left|\nu_{0}\right|^{2}\mu_{0}\,dxdt$
$\displaystyle\leq\frac{1}{\beta}\int_{0}^{\delta_{0}^{\beta}}\\!\\!\frac{1}{t^{\frac{1}{\beta}}}\,dt$

which is finite for every $\beta>1$. With such a choice, if we merge this
estimate with (2.9) we obtain

$\int_{0}^{\delta_{0}}\\!\\!\int_{M}\left|\tilde{\nu}_{0}\right|^{2}\tilde{\mu}_{0}\,dxdt+\varepsilon\int_{0}^{\delta_{0}}\\!\int_{M}\tilde{\mu}_{0}(t)(\log\tilde{\mu}_{0}(t)+V)\,dxdt\leq
C_{0}$

where $C_{0}$ is a constant depending only on $M,d,\lambda,T,\|\varepsilon
V\|_{\infty},\beta$ and $\delta_{0}$.
In a similar way, for a fixed $T/3<\delta_{1}<T$, we find a smooth curve of
probability densities $\tilde{\mu}_{1}$, with velocity $\tilde{\nu}_{1}$ such
that

$\int_{\delta_{1}}^{T}\\!\\!\int_{M}\left|\tilde{\nu}_{1}\right|^{2}\tilde{\mu}_{1}\,dxdt+\varepsilon\int_{\delta_{1}}^{T}\\!\int_{M}\tilde{\mu}_{1}(t)(\log\tilde{\mu}_{1}(t)+V)\,dxdt\leq
C_{1}$

where $C_{1}$ is a constant depending only on $M,d,\lambda,T,\|\varepsilon
V\|_{\infty},\beta$ and $\delta_{1}$.
Consider now the 2-Wasserstein geodesic $(\overline{\mu},\overline{\nu})$
between $\tilde{\mu}_{0}(\delta_{0},\cdot)$ and
$\tilde{\mu}_{1}(\delta_{1},\cdot)$. We recall (see e.g. [15]) that the
entropy functional is $(-\lambda)$-convex along the 2-Wasserstein geodesic.
Hence, by (2.9),

$\int_{\delta_{0}}^{\delta_{1}}\\!\\!\int_{M}\left|\overline{\nu}\right|^{2}\overline{\mu}\,dxdt+\varepsilon\int_{\delta_{0}}^{\delta_{1}}\\!\\!\int_{M}\overline{\mu}(\log\overline{\mu}+V)\,dxdt\leq
C$

where the constant $C$ depends only on $M,d,\lambda,T,\|\varepsilon
V\|_{\infty},\delta_{0},\delta_{1}$ and the Wasserstein distance
$W_{2}\bigl{(}\tilde{\mu}_{0}(\delta_{0},\cdot),\tilde{\mu}_{1}(\delta_{1},\cdot)\bigr{)}$.
However, the latter is uniformly estimated in terms of the manifold, thanks to
the compactness of $M$. Finally, gluing the paths from $m_{0}$ to
$\tilde{\mu}_{0}(\delta_{0},\cdot)$ and from
$\tilde{\mu}_{1}(\delta_{1},\cdot)$ to $m_{1}$ with the 2-Wasserstein geodesic
$\overline{\mu}$, we have built an admissible arc joining $m_{0}$ and $m_{1}$,
and with a convenient choice of $\delta_{0},\delta_{1}$ we estimate

$\inf\mathcal{F}_{\varepsilon}(m_{0},m_{1})\leq C_{0}+C+C_{1}$

where last constants only depend on $M,d,\lambda,T,\|\varepsilon
V\|_{\infty}$.

It is a classical result, after Benamou-Brenier’s trick [2] and the weak-lower
semicontinuity of the entropy, that the above estimate, with the existence of
an admissible curve, yields the existence of a minimizer of
$\mathcal{F}_{\varepsilon}$ by direct methods of Calculus of Variations. For a
similar proof in euclidean context, see e.g. [27, Proposition 2.9]. ∎

######  Remark 2.3:

With a suitable choice of $\delta_{0},\delta_{1}$ in the above proof, it is
possible to give an estimate of the dependence of the constant $C$ in (2.6)
from the time-horizon $T$. Alternatively, if we denote
$\min\mathcal{F}_{\varepsilon}^{1}$ the minimum in unit time $T=1$, with a
simple time-scaling one can estimate

$\min\mathcal{F}_{\varepsilon}\leq\max\left(\frac{1}{T},T\right)\min\mathcal{F}_{\varepsilon}^{1}\leq\max\left(\frac{1}{T},T\right)C(M,d,\lambda)$

######  Remark 2.4:

We stress that in the above proof we can avoid the use of the Wasserstein
geodesic $(\overline{\mu},\overline{\nu})$ between
$\tilde{\mu}_{0}(\delta_{0},\cdot)$ and $\tilde{\mu}_{1}(\delta_{1},\cdot)$
(and consequently, avoid the use of the $(-\lambda)$\- displacement convexity
of the geodesic). In fact, since $\tilde{\mu}_{0}(\delta_{0},\cdot)$ and
$\tilde{\mu}_{1}(\delta_{1},\cdot)$ are smooth and positive, we can take the
smooth optimal curve of $\mathcal{F}_{\varepsilon}$ joining the two measures,
whose existence will be proved in Section 5, Theorem 5.4.

##  3 The optimality system

In this Section we discuss the structure of the optimality system satisfied by
minima of functional (2.5). This is a first order PDE system which takes the
following form

$\left\\{\begin{aligned} &-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}=\varepsilon(\log(m)+V)\,,\qquad t\in(0,T)\\\
&\partial_{t}m-div_{g}(m\nabla u)=0\,,\qquad\qquad\qquad
t\in(0,T).\end{aligned}\right.$ (3.1)

Let us first underline that, at least formally, (3.1) is the optimality
condition of (2.5), in the sense that $(m,\nabla u)$ provides with the couple
$(m,v)$ minimizing (2.5). This is a consequence of the convexity of the
functional, following the original idea of Benamou and Brenier [2]. Even if
this is clear to expert readers, we provide a proof for completeness.

######  Lemma 3.1:

Let $(u,m)$ be a smooth solution of system (3.1), in the sense that $u\in
C^{1}([0,T]\times M),m\in C^{0}([0,T]\times M)$ with $m(0)=m_{0},m(T)=m_{1}$,
and $m>0$. Then $(m,\nabla u)$ is a minimum point of (2.5).

###### Proof.

Let $(\mu,v)$ be any couple which solves the continuity equation in the sense
of Definition 2.1. We can assume that
$\mathcal{F}_{\varepsilon}(\mu,v)<\infty$ (otherwise, the inequality
$\mathcal{F}_{\varepsilon}(m,\nabla u)\leq\mathcal{F}_{\varepsilon}(\mu,v)$ is
obvious), and in particular $\mu\in L^{1}(Q_{T})$. Let us define the following
convex and lower semicontinuous function in $\mathbb{R}^{d}\times\mathbb{R}$:

$\Psi(p,m)=\left\\{\begin{array}[c]{ll}\frac{|p|^{2}}{2m}&\hbox{if}\quad
m>0,\\\ 0&\hbox{if}\quad m=0\hbox{ and }p=0,\\\
+\infty&\hbox{otherwise}.\end{array}\right.$ (3.2)

We set $w=\mu v,\hat{w}=m\nabla u$. Since $m,u$ are smooth solutions of (3.1)
(with $m>0$), then $\partial\Psi(\hat{w},m)$ is well defined. By convexity of
$\Psi$, we have

$\displaystyle\Psi(\hat{w},m)-\Psi(w,\mu)$
$\displaystyle\leq\partial_{p}\Psi(\hat{w},m)\cdot(\hat{w}-w)+\partial_{m}\Psi(\hat{w},m)(m-\mu)$
$\displaystyle=\frac{\hat{w}}{m}\cdot(\hat{w}-w)-\frac{|\hat{w}|^{2}}{2m^{2}}(m-\mu)$

Since $\mu\in L^{1}$ and $\mu|v|^{2}\in L^{1}$, we have $w\in L^{1}$ and the
above inequality is integrable on $M$. From the very definition of $w,\hat{w}$
we deduce

$\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m|\nabla
u|^{2}\leq\frac{1}{2}\int_{0}^{T}\\!\\!\\!\\!\int_{M}\mu|v|^{2}+\frac{1}{2}\int_{0}^{T}\\!\\!\\!\\!\int_{M}m|\nabla
u|^{2}-\int_{0}^{T}\\!\\!\\!\\!\int_{M}w\cdot\nabla
u+\frac{1}{2}\int_{0}^{T}\\!\\!\\!\\!\int_{M}\mu|\nabla u|^{2}$ (3.3)

Since $u\in C^{1}$, the continuity equation gives

$\displaystyle\int_{0}^{T}\\!\\!\\!\\!\int_{M}w\cdot\nabla u$
$\displaystyle=\int_{0}^{T}\\!\\!\\!\\!\int_{M}\partial_{t}u\,\mu-\int_{M}u(T)m_{1}+\int_{M}u(0)m_{0}$
$\displaystyle=-\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}(\log
m+V)\mu+\frac{1}{2}|\nabla u|^{2}\mu-\int_{M}u(T)m_{1}+\int_{M}u(0)m_{0}\,.$

Hence from (3.3) we get

$\displaystyle\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m|\nabla u|^{2}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{T}\\!\\!\\!\\!\int_{M}\mu|v|^{2}+\frac{1}{2}\int_{0}^{T}\\!\\!\\!\\!\int_{M}m|\nabla
u|^{2}+\int_{M}u(T)m_{1}-\int_{M}u(0)m_{0}$
$\displaystyle+\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}(\log m+V)\mu$
$\displaystyle=\frac{1}{2}\int_{0}^{T}\\!\\!\\!\\!\int_{M}\mu|v|^{2}-\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}m(\log(m)+V)+\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}(\log
m+V)\mu$

By convexity we obviously have
$m\log(m)-\mu\log(\mu)\leq\log(m)(m-\mu)+(m-\mu)$, where last term disappears
after integration. Then we conclude that

$\mathcal{F}_{\varepsilon}(m,\nabla u)\leq\mathcal{F}_{\varepsilon}(\mu,v)\,.$

∎

Since (3.1) is a Hamiltonian system, there is some invariant of motion. The
proof is straightforward.

######  Lemma 3.2:

Let $(u,m)$ be a smooth solution of system (3.1). Then we have that the
quantity

$E(m_{0},m_{1}):=\frac{1}{2}\int_{M}m|\nabla
u|^{2}-\varepsilon\int_{M}m(\log(m)+V)$

is constant in time and it holds

$E(m_{0},m_{1})=\frac{1}{T}{\mathcal{B}}_{\varepsilon}(m_{0},m_{1})-\frac{2\varepsilon}{T}\int_{0}^{T}\\!\\!\\!\\!\int_{M}m(\log(m)+V),$
(3.4)

where
${\mathcal{B}}_{\varepsilon}(m_{0},m_{1})=\min\mathcal{F}_{\varepsilon}$.

We observe that the quantity $E$ only depends on the marginals $m_{0},m_{1}$,
and is easily estimated. In particular, by Jensen’s inequality, we have
$\int_{M}m(\log m+V)dx\geq-\log(\nu(M))$, for the measure $\nu:=e^{-V}dx$.
Recalling Proposition 2.2 (and Remark 2.3), we deduce that

$E(m_{0},m_{1})\leq\frac{1}{T^{2}\wedge
1}\,C(M,d,\lambda)+2\varepsilon\log(\int_{M}e^{-V}dx)\leq K$ (3.5)

for some constant $K$ which is uniform for all
$m_{0},m_{1}\in{\mathcal{P}}(M),V\in L^{\infty}(M)$ and any $\varepsilon\leq
1$.

###  3.1 Displacement convexity estimates

In this section we study the convexity of some energy functional along the
optimal curves of (2.5). This is obtained in the Eulerian approach by
exploiting dissipativity properties of the solutions of system (3.1). We
consider smooth solutions, which justifies the computations below; as we will
see later, this is no loss of generality, since all solutions will be obtained
as limit of classical ones. The following is an extension to the Riemannian
setting of the results proved in [19], (or in [43] with Neumann conditions);
the only new ingredient is provided by the Böchner formula (2.1).

######  Proposition 3.3:

Let $u\in C^{2}(\overline{Q}_{T})$ and $m\in C^{1}(\overline{Q}_{T})$ be
classical solutions to the system (3.1), where $V\in W^{2,\infty}(M)$.

Let $U:(0,+\infty)\rightarrow\mathbb{R}$ be a $C^{1}$ function such that

$P(r)\coloneqq U^{\prime}(r)r-U(r)\geq 0$

Then

$\begin{split}\frac{d^{2}}{dt^{2}}\int_{M}U(m)\,dx\geq&\int_{M}\left[P^{\prime}(m)m-(1-\frac{1}{d})P(m)\right](\Delta_{g}u)^{2}\,dx\\\
&+\int_{M}P(m)Ricc_{g}(\nabla u,\nabla u)\,dx+\\\
&+\int_{M}\varepsilon\frac{P^{\prime}(m)}{m}\left|\nabla
m\right|^{2}\,dx+\varepsilon\int_{M}P^{\prime}(m)\nabla
m\operatorname{\cdot_{g}\\!}\nabla V\,dx\end{split}$ (3.6)

###### Proof.

We begin by calculating the first derivate of the function
$t\rightarrow\int_{M}U(m)\,dx$,

$\displaystyle\frac{d}{dt}\int_{M}U(m)\,dx$
$\displaystyle=\int_{M}U^{\prime}(m)\partial_{t}m\,dx$
$\displaystyle=\int_{M}U^{\prime}(m)(m\Delta_{g}u+\nabla
m\operatorname{\cdot_{g}\\!}\nabla u)\,dx$
$\displaystyle=\int_{M}P(m)\Delta_{g}u\,dx$

recalling that $P(r)=U^{\prime}(r)r-U(r)$. So, the second derivative takes the
form

$\displaystyle\frac{d^{2}}{dt^{2}}\int_{M}U(m)\,dx=$
$\displaystyle\int_{M}P^{\prime}(m)\partial_{t}m\Delta_{g}u+P(m)\Delta_{g}(\partial_{t}u)\,dx$
$\displaystyle=$ $\displaystyle\int_{M}P^{\prime}(m)(m\Delta_{g}u+\nabla
m\operatorname{\cdot_{g}\\!}\nabla u)\Delta_{g}u\,dx+$
$\displaystyle+\int_{M}P(m)\Delta_{g}(\frac{1}{2}\left|\nabla
u\right|^{2}-\varepsilon(\log(m)+V))\,dx$ $\displaystyle=$
$\displaystyle\int_{M}[P^{\prime}(m)m-P(m)](\Delta_{g}u)^{2}\,dx-\int_{M}P(m)\nabla(\Delta_{g}u)\operatorname{\cdot_{g}\\!}\nabla
u\,dx$ $\displaystyle+\int_{M}P(m)\Delta_{g}(\frac{1}{2}\left|\nabla
u\right|^{2}-\varepsilon(\log(m)+V))\,dx\,.$

Now we use Böchner’s formula (2.1) to calculate
$\Delta_{g}(\frac{1}{2}\left|\nabla u\right|^{2})$, obtaining

$\displaystyle\frac{d^{2}}{dt^{2}}\int_{M}U(m)\,dx=$
$\displaystyle\int_{M}[P^{\prime}(m)m-P(m)](\Delta_{g}u)^{2}\,dx$
$\displaystyle+\int_{M}P(m)[tr((\nabla^{2}u)^{2})+Ricc_{g}(\nabla u,\nabla
u)]\,dx$ $\displaystyle+\int_{M}\varepsilon\frac{P^{\prime}(m)}{m}\left|\nabla
m\right|^{2}dx+\varepsilon\int_{M}P^{\prime}(m)\nabla
m\operatorname{\cdot_{g}\\!}\nabla V\,dx.$

Since, by the symmetry of $\nabla^{2}u$, it holds
$tr((\nabla^{2}u)^{2})\geq\frac{1}{d}(tr(\nabla^{2}u))^{2}=\frac{1}{d}(\Delta_{g}u)^{2}$,
we get (3.6). ∎

In particular, the inequality (3.6) implies the semi-convexity of the
$\log$-entropy along the optimal curve $m(t)$, and the strict convexity of the
relative entropy whenever $Ricc_{g}+D^{2}V\geq 0$.

######  Corollary 3.4:

Under the assumptions of Proposition 3.3, let ${\mathcal{H}}(m(t);\nu)$ be the
relative entropy defined in (2.4), for $\nu=e^{-V}dx$. Then we have

$\begin{split}\frac{d^{2}}{dt^{2}}{\mathcal{H}}(m(t);\nu)&\geq\int_{M}m(Ricc_{g}(\nabla
u,\nabla u)+D^{2}V(\nabla u,\nabla u))\,dx\\\
&\qquad\qquad+\varepsilon\int_{M}|\nabla(\log m+V)|^{2}\,m\,dx\,.\end{split}$
(3.7)

Moreover, let $\lambda\geq 0$ satisfy (2.2), and define

$\varphi(t)=\int_{M}m(t)\log m(t)\,dx\,.$

Then we have:

  * (i)

there exists a constant $\Lambda_{\varepsilon}$, depending on
$M,d,\lambda,\|V\|_{W^{1,\infty}},T,\varepsilon$, such that $\varphi$ is
$\Lambda_{\varepsilon}-$ semiconvex in $(0,T)$, hence

$\displaystyle\varphi(t)\leq\frac{T-t}{T}\varphi(0)+\frac{t}{T}\varphi(T)+\Lambda_{\varepsilon}\frac{t(T-t)}{2T^{2}}$
(3.8)

Moreover, the constant $\Lambda_{\varepsilon}$ is bounded independently of
$\varepsilon$ (for $\varepsilon\leq 1$), and we have
$\Lambda_{\varepsilon}\,\mathop{\to}\limits^{\varepsilon\to
0}\,\frac{\lambda}{T}W_{2}(m_{0},m_{1})^{2}$.

  * (ii)

there exists a constant $L=L(M,d,\lambda,\|V\|_{W^{1,\infty}},T)$ such that

$\varphi(t)\leq
d\,|\log(t(T-t))|+\frac{d}{2}\,|\log\varepsilon|+L\qquad\forall t\in(0,T)\,.$
(3.9)

###### Proof.

We use Proposition 3.3 with $U(r)=r\log r-r$ (so that $P(r)=r$) and we get

$\begin{split}\frac{d^{2}}{dt^{2}}\int_{M}m\log
m=&\frac{d^{2}}{dt^{2}}\int_{M}(m\log m-m)\,dx\geq\int_{M}mRicc_{g}(\nabla
u,\nabla u)\,dx+\\\ &\qquad+\int_{M}\varepsilon\frac{1}{m}\left|\nabla
m\right|^{2}\,dx+\varepsilon\int_{M}\nabla m\operatorname{\cdot_{g}\\!}\nabla
V\,dx\,.\end{split}$ (3.10)

Similarly, we compute

$\displaystyle\frac{d^{2}}{dt^{2}}\int_{M}m\,V=$
$\displaystyle\frac{d}{dt}\int_{M}\partial_{t}m\,V=-\frac{d}{dt}\int_{M}m\,\nabla
V\operatorname{\cdot_{g}\\!}\nabla u$ $\displaystyle=-\int_{M}(\nabla
V\operatorname{\cdot_{g}\\!}\nabla u)div_{g}(m\nabla u)-\int_{M}m\,\nabla
V\operatorname{\cdot_{g}\\!}\nabla(\frac{1}{2}|\nabla
u|^{2}-\varepsilon(log(m)+V))$ $\displaystyle=\int_{M}m\,D^{2}V(\nabla
u,\nabla u)+\varepsilon\int_{M}m\,\nabla
V\operatorname{\cdot_{g}\\!}\nabla(log(m)+V)\,.$

Adding this equality to (3.10), we obtain (3.7).

Now, if we come back to (3.10) and use the lower bound on $Ric_{g}$, we get

$\frac{d^{2}}{dt^{2}}\int_{M}m\log m\geq-\lambda\int_{M}m\left|\nabla
u\right|^{2}\,dx+\frac{\varepsilon}{2}\int_{M}\frac{1}{m}\left|\nabla
m\right|^{2}\,dx-\frac{\varepsilon}{2}\int_{M}m|\nabla V|^{2}\,dx\,.$

By definition of the quantity $E(m_{0},m_{1})$ we obtain

$\begin{split}\frac{d^{2}}{dt^{2}}\int_{M}m\log m\geq&-2\lambda
E(m_{0},m_{1})-2\lambda\varepsilon\int_{M}m(\log m+V)dx\\\
&+\frac{\varepsilon}{2}\int_{M}\frac{1}{m}\left|\nabla
m\right|^{2}\,dx-\frac{\varepsilon}{2}\int_{M}m|\nabla V|^{2}\,dx\\\
\geq&-2\lambda E(m_{0},m_{1})-2\lambda\varepsilon\int_{M}m\log mdx\\\
&+\frac{\varepsilon}{2}\int_{M}\frac{1}{m}\left|\nabla
m\right|^{2}\,dx-\varepsilon\,c(\lambda,\|V\|_{W^{1,\infty}}).\end{split}$
(3.11)

We estimate the Fisher information of $m$ as in Proposition 2.2, see (2.8):

$\int_{M}\frac{1}{m}\left|\nabla m\right|^{2}\,dx\geq
c_{1}\exp\left(\frac{2}{d}\int_{M}m\log m\right)-c_{2}Vol(M)\,.$

Therefore, if $\varphi(t)\coloneqq\int_{M}m\log m\,dx$, we deduce

$\varphi^{\prime\prime}\geq-2\lambda
E(m_{0},m_{1})+\varepsilon(-2\lambda\varphi+c_{3}e^{\frac{2}{d}\varphi})-\varepsilon\,c(\lambda,\|V\|_{W^{1,\infty}},M)\,,$
(3.12)

for some constant $c_{3}$. We note that the function $r\to-2\lambda
r+c_{3}e^{\frac{2}{d}r}$ has a finite minimum on $[0,+\infty)$, and that
$E(m_{0},m_{1})$ is bounded above by some constant $K$ only depending on
$M,d,\lambda,T,\|V\|_{\infty}$, see (3.5). Hence we have

$\varphi^{\prime\prime}\geq-2\lambda K-\varepsilon
C(\lambda,\|V\|_{W^{1,\infty}},M)$

which gives the semiconvexity of the entropy along the optimal curves, with a
semi-convexity constant $\Lambda_{\varepsilon}$ which is bounded uniformly for
$\varepsilon\leq 1$. In a more precise form, on account of (3.4) we can
estimate

$\Lambda_{\varepsilon}=\varepsilon C+2\lambda E(m_{0},m_{1})\simeq\varepsilon
C(1+|\log\varepsilon|)+2\frac{\lambda}{T}\min(\mathcal{F}_{\varepsilon})\,.$

As we will prove in Section 6, it holds that
$\min(\mathcal{F}_{\varepsilon})\to\frac{1}{2}W_{2}(m_{0},m_{1})^{2}$; hence
we deduce

$\Lambda_{\varepsilon}\,\mathop{\to}^{\varepsilon\to
0}\,\,\frac{\lambda}{T}W_{2}(m_{0},m_{1})^{2}\,.$

Now we also obtain a local bound for the entropy, independently from the
initial and terminal marginals. Indeed, we deduce from (3.12) that

$\varphi^{\prime\prime}\geq\varepsilon\,c_{4}e^{\frac{2}{d}\varphi}-c_{5}\qquad
t\in(0,T),$

for some constant $c_{4},c_{5}$ depending on
$M,d,\lambda,\|V\|_{W^{1,\infty}},T,\varepsilon$ (and uniform for
$\varepsilon\leq 1$). With a suitable choice of $L$ (depending on
$c_{4},c_{5},T$), we have that the function

$\psi(t):=-d\log(\sqrt{\varepsilon}\,t(T-t))+L$

is a supersolution of the same equation, i.e.
$\psi^{\prime\prime}\leq\varepsilon\,c_{4}e^{\frac{2}{d}\psi}-c_{5}$ for
$t\in(0,T)$. Since $\psi$ blows-up at $t=0,t=T$, we conclude by comparison
that $\varphi\leq\psi$, which gives (3.9). ∎

###  3.2 The optimality system as an elliptic equation

System (3.1) can be recasted as a single elliptic equation, in time-space
variables, for $u$. This comes by noting that
$me^{V}=\exp(\frac{1}{\varepsilon}(\frac{|\nabla u|^{2}}{2}-\partial_{t}u))$,
which can be inserted in the continuity equation, giving rise to a quasilinear
elliptic equation in divergence form for $u$.

This is in fact a special case of a general approach suggested by P-L. Lions
in his lectures at Collège de France [35], in order to handle mean-field game
systems of first order, such as

$\left\\{\begin{aligned} &-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}=f(m)+V\\\ &\partial_{t}m-m\Delta_{g}u-\nabla
u\operatorname{\cdot_{g}\\!}\nabla m=0\end{aligned}\right.$ (3.13)

whenever $f$ is an increasing function. For the reader’s convenience, we
derive here this equation in the Riemannian context. To this purpose, we first
compute the covariant gradient of both terms in the Hamilton-Jacobi equation:

$\begin{split}f^{\prime}(m)\nabla
m&=\nabla\left(-\partial_{t}u+\frac{\left|\nabla
u\right|^{2}}{2}-V\right)\,.\end{split}$

Taking the scalar product with $\nabla u$ we get

$f^{\prime}(m)\nabla m\operatorname{\cdot_{g}\\!}\nabla
u=\nabla\left(-\partial_{t}u+\frac{\left|\nabla
u\right|^{2}}{2}-V\right)\operatorname{\cdot_{g}\\!}\nabla u\,.$

Similarly, by taking the time derivative from the Hamilton-Jacobi equation, we
have:

$\begin{split}f^{\prime}(m)\partial_{t}m&=\frac{\partial}{\partial
t}\left(-\partial_{t}u+\frac{\left|\nabla u\right|^{2}}{2}-V\right)\\\
&=-\partial_{tt}u+\nabla(\partial_{t}u)\operatorname{\cdot_{g}\\!}\nabla
u\,.\\\ \end{split}$ (3.14)

From the above equalities, merged with the continuity equation for $m$, we
obtain

$\begin{split}-\partial_{tt}u+\nabla(\partial_{t}u)\operatorname{\cdot_{g}\\!}\nabla
u-\nabla\left(-\partial_{t}u+\frac{\left|\nabla
u\right|^{2}}{2}-V\right)\operatorname{\cdot_{g}\\!}\nabla
u&=f^{\prime}(m)\left(\partial_{t}m-\nabla m\operatorname{\cdot_{g}\\!}\nabla
u\right)\\\ &=f^{\prime}(m)\,m\Delta_{g}u\end{split}$ (3.15)

which becomes a second order equation in the only unknown $u$. Indeed, by
setting $\phi\coloneqq(f)^{-1}$, the first equation of the system reads as

$m=\phi\left(-\partial_{t}u+\frac{\left|\nabla u\right|^{2}}{2}-V\right)\,,$

and so (3.15) becomes a single equation for $u$. In the particular case of
$f(m)=\varepsilon\log m$, we have $f^{\prime}(m)m=\varepsilon$, and (3.15)
simplifies further; if we also scale the potential $V$ into $\varepsilon\,V$
(this is not necessary of course, but it is more consistent with the model of
relative entropy), we get to the following form:

$-\partial_{tt}u+2\nabla
u\operatorname{\cdot_{g}\\!}\nabla(\partial_{t}u)-(\nabla^{2}u)(\nabla
u,\nabla u)-\varepsilon\Delta_{g}u+\varepsilon\nabla
u\operatorname{\cdot_{g}\\!}\nabla V=0.$ (3.16)

We can shorten such equation by writing it as a quasilinear equation in the
space-time variable

$-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}u\right)+\varepsilon\nabla
u\operatorname{\cdot_{g}\\!}\nabla V(x)=0$ (3.17)

where $\mathcal{A}(x,\eta)$ is the endomorphism of $\mathbb{R}\times
T_{x}\\!M$ defined by

$\begin{array}[]{ccl}\mathbb{R}\times
T_{x}\\!M&\longrightarrow&\mathbb{R}\times T_{x}\\!M\\\
[\overline{w},w]&\longrightarrow&[-(\eta\operatorname{\cdot_{g}\\!}w-\overline{w}),(\eta\operatorname{\cdot_{g}\\!}w-\overline{w})\eta+\varepsilon
w]\end{array}$

and, for every $C^{2}$ function $f$ on $[0,T]\times M$, the endomorphism
$\overline{\nabla}^{2}f$ is given by

$\begin{array}[]{ccl}\mathbb{R}\times
T_{x}\\!M&\longrightarrow&\mathbb{R}\times T_{x}\\!M\\\
[\overline{w},w]&\longrightarrow&[\overline{w}\partial_{tt}f+\nabla(\partial_{t}f)\operatorname{\cdot_{g}\\!}w,\overline{w}\nabla(\partial_{t}f)+D_{w}\nabla
f]\end{array}$

Note that $\mathcal{A}(x,\eta)$ is independent of $t\in[0,T]$ and it is
symmetric for every choice $(x,\eta)\in M\times T_{x}M$. The symbol
$\mathcal{A}(x,\eta)$ will denote also the bilinear form induced by such
endomorphism through the product metric $\underline{g}$ of the manifold
$\mathbb{R}\times M$. Namely,

$\displaystyle\mathcal{A}(x,\eta)([\overline{w},w],[\overline{v},v])$
$\displaystyle\coloneqq\left(\mathcal{A}(x,\eta)[\overline{w},w]\right)\cdot_{\underline{g}}[\overline{v},v]$
$\displaystyle=(\eta\operatorname{\cdot_{g}\\!}w-\overline{w})(\eta\operatorname{\cdot_{g}\\!}v-\overline{v})+\varepsilon
w\operatorname{\cdot_{g}\\!}v\,.$

Finally we note that such bilinear form is elliptic (though not uniformly), in
fact for every $[\overline{w},w]\in\mathbb{R}\times T_{x}\\!M$ we have

$\mathcal{A}(x,\eta)([\overline{w},w],[\overline{w},w])=(\eta\operatorname{\cdot_{g}\\!}w-\overline{w})^{2}+\varepsilon\left|w\right|^{2}>0\qquad\forall[\overline{w},w]\neq[0,0].$

##  4 Gradient bounds for smooth solutions

In this section we obtain estimates for smooth solutions of the system (1.2),
by exploiting the elliptic character of the quasilinear equation (3.17). We
mostly follow the ideas developed in [35], [40], [43], although specifying to
the case of the entropy nonlinearity allows us to simplify some argument and
to give estimates in a more precise form.

As a first step, we derive from (3.16) the differential equations solved by
some auxiliary functions of $u$ and its derivatives.

######  Lemma 4.1:

Let $u\in C^{3}([0,T]\times M)$ be a solution of

$-tr\left(\mathcal{A}(x,\nabla u)\circ\overline{\nabla}^{2}u\right)+\rho
u+\varepsilon\nabla u\operatorname{\cdot_{g}\\!}\nabla V(x)=0$ (4.1)

Then

  1. i)

for every $K\in\mathbb{R}$, the function $h\coloneqq(u+K)^{2}$ satisfies

$\displaystyle-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}h\right)+\varepsilon\nabla
u\operatorname{\cdot_{g}\\!}\nabla V$ $\displaystyle+2\rho u\,(u+K)$
$\displaystyle=-2\mathcal{A}(x,\nabla u)\left([\partial_{t}u,\nabla
u],[\partial_{t}u,\nabla u]\right).$

  2. ii)

Set $\omega\coloneqq\partial_{t}u$, then it satisfies

$\displaystyle-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}\omega\right)+\varepsilon\nabla\omega\operatorname{\cdot_{g}\\!}\nabla
V+\rho\omega=$
$\displaystyle-2\left|\nabla\omega\right|^{2}+2(\nabla^{2}u)(\nabla
u,\nabla\omega).$

  3. iii)

Set $\varphi\coloneqq\frac{1}{2}\left|\nabla u\right|^{2}$, then it satisfies

$\displaystyle-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}\varphi\right)$
$\displaystyle+\varepsilon\nabla\varphi\operatorname{\cdot_{g}\\!}\nabla
V+2\rho\varphi=\left|\nabla\varphi\right|^{2}-|\nabla(\partial_{t}u)|^{2}$
$\displaystyle-\varepsilon\left(\left|\nabla^{2}u\right|^{2}+(\nabla^{2}V)(\nabla
u,\nabla u)+Ricc_{g}(\nabla u,\nabla u)\right)$

###### Proof.

Equations i) and ii) are straightful computations based on equation (4.1). For
i) we use the chain rule. For ii) we derive in time equation (4.1) and we use
that $\mathcal{A}(x,\nabla u)$ is independent from time.

iii) We now study the differential equation solved by
$\varphi\coloneqq\tfrac{1}{2}\left|\nabla u\right|^{2}$. We first observe
that, if we develop the trace in (4.1), as we did in (3.16), we can rewrite
the equation as

$\partial_{tt}u+\varepsilon\Delta_{g}u=2\partial_{t}\varphi-\nabla
u\operatorname{\cdot_{g}\\!}\nabla\varphi+\varepsilon\nabla
u\operatorname{\cdot_{g}\\!}\nabla V+\rho u$ (4.2)

Now, using the Bochner’s formula (2.1), which means

$\Delta_{g}\varphi=\tfrac{1}{2}\Delta_{g}\left|\nabla
u\right|^{2}=\left|\nabla^{2}u\right|^{2}+\nabla(\Delta_{g}u)\operatorname{\cdot_{g}\\!}\nabla
u+Ricc_{g}(\nabla u,\nabla u)$ (4.3)

we get

$\displaystyle\partial_{tt}\varphi+\varepsilon\Delta_{g}\varphi$
$\displaystyle=\nabla(\partial_{t}u)\operatorname{\cdot_{g}\\!}\nabla(\partial_{t}u)+\nabla
u\operatorname{\cdot_{g}\\!}\nabla(\partial_{tt}u)+\varepsilon\left(\nabla(\Delta_{g}u)\operatorname{\cdot_{g}\\!}\nabla
u+\left|\nabla^{2}u\right|^{2}+Ricc_{g}(\nabla u,\nabla u)\right)$
$\displaystyle=\nabla\bigl{(}\partial_{tt}u+\varepsilon\Delta_{g}u\bigr{)}\operatorname{\cdot_{g}\\!}\nabla
u+|\nabla(\partial_{t}u)|^{2}+\varepsilon\left[\left|\nabla^{2}u\right|^{2}+Ricc_{g}(\nabla
u,\nabla u)\right]\,.$

Then, using (4.2), we have

$\displaystyle\partial_{tt}\varphi+\varepsilon\Delta_{g}\varphi$
$\displaystyle=2\nabla(\partial_{t}\varphi)\operatorname{\cdot_{g}\\!}\nabla
u-(\nabla^{2}u)(\nabla u,\nabla\varphi)-(\nabla^{2}\varphi)(\nabla u,\nabla
u)+$ $\displaystyle\quad+\varepsilon(\nabla^{2}u)(\nabla u,\nabla
V)+\varepsilon(\nabla^{2}V)(\nabla u,\nabla u)+2\rho\varphi+$
$\displaystyle\quad+|\nabla(\partial_{t}u)|^{2}+\varepsilon\left[\left|\nabla^{2}u\right|^{2}+Ricc_{g}(\nabla
u,\nabla u)\right]\,.$

We note that for every vector field $v$ on $M$ we have
$\nabla\varphi\operatorname{\cdot_{g}\\!}v=(\nabla^{2}u)(\nabla u,v)$, so the
above equality becomes

$\displaystyle\partial_{tt}\varphi+\varepsilon\Delta_{g}\varphi$
$\displaystyle=2\nabla(\partial_{t}\varphi)\operatorname{\cdot_{g}\\!}\nabla
u-\nabla\varphi\operatorname{\cdot_{g}\\!}\nabla\varphi-(\nabla^{2}\varphi)(\nabla
u,\nabla u)+\varepsilon\nabla\varphi\operatorname{\cdot_{g}\\!}\nabla V+$
$\displaystyle+\varepsilon(\nabla^{2}V)(\nabla u,\nabla
u)+2\rho\varphi+|\nabla(\partial_{t}u)|^{2}+$
$\displaystyle+\varepsilon\left[\left|\nabla^{2}u\right|^{2}+Ricc_{g}(\nabla
u,\nabla u)\right].$

Finally, if we look at the whole chain of equalities we get

$\displaystyle-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}\varphi\right)+\varepsilon\nabla\varphi\operatorname{\cdot_{g}\\!}\nabla
V+2\rho\varphi=\left|\nabla\varphi\right|^{2}-|\nabla(\partial_{t}u)|^{2}$
$\displaystyle\qquad\qquad\qquad-\varepsilon\left(\left|\nabla^{2}u\right|^{2}+(\nabla^{2}V)(\nabla
u,\nabla u)+Ricc_{g}(\nabla u,\nabla u)\right)\,.$

∎

###  4.1 Global Lipschitz bound on $u$

In this section we will prove a $W^{1,\infty}$ bound for the solution $u\in
C^{3}(Q_{T})$ of the system

$\begin{cases}-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}u\right)+\rho u+\varepsilon\nabla
u\operatorname{\cdot_{g}\\!}\nabla V(x)=0&\text{in $Q_{T}$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}=\delta
u+\varepsilon(\log(m_{1})+V(x))&\text{in $t=T,x\in M$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}+\delta
u=\varepsilon(\log(m_{0})+V(x))&\text{in $t=0,x\in M$.}\\\ \end{cases}$ (4.4)

We recall that $Q_{T}=(0,T)\times M$, and hereafter we denote

$\displaystyle\Sigma_{0}\coloneqq\\{0\\}\times
M\quad\Sigma_{T}\coloneqq\\{T\\}\times M.$

We notice that (4.4) is a perturbation of (3.17), containing an additional
term $\rho u$ in the interior (this will simplify a preliminary step, detailed
in Appendix) and an additional term $\delta u$ on the boundary. The latter is
used to control the function $u$ in a first step. Indeed, using the maximum
principle, we have

$\delta\lVert
u\rVert_{\infty}\leq\left(\lVert\varepsilon(\log(m_{0})+V)\rVert_{\infty}+\lVert\varepsilon(\log(m_{1})+V)\rVert_{\infty}\right).$
(4.5)

Analogously, using Lemma 4.1 and the maximum principle, we bound the time
derivative.

######  Lemma 4.2:

Let $u$ be a solution of (4.4). It holds that

$\partial_{t}u(t,x)\leq\sup_{\Sigma_{0}\cup\Sigma_{T}}(\partial_{t}u)_{+}\quad\text{and}\quad\left|\partial_{t}u(t,x)\right|\leq\sup_{\Sigma_{0}\cup\Sigma_{T}}\left|\partial_{t}u(t,x)\right|\quad\forall(t,x)\in
Q_{T}$ (4.6)

Finally we give a bound for the space derivative in terms of the $\sup$-norm
of $u$.

######  Theorem 4.3:

Let $u$ be a solution of (4.4). There exists a constant $C$, independent from
$\rho$ and $\delta$, such that

$\lVert{\nabla}u\rVert_{\infty}\leq C(1+\lVert
u\rVert_{\infty})\qquad;\qquad\lVert{\partial_{t}}u\rVert_{\infty}\leq
C(1+\lVert u\rVert_{\infty}^{2}).$ (4.7)

Such constant $C$ depends on
$\lVert\varepsilon\log(m_{0})\rVert_{W^{1,\infty}},\lVert\varepsilon\log(m_{1})\rVert_{W^{1,\infty}}$
and on $\lVert\varepsilon V\rVert_{W^{2,\infty}}$.

###### Proof.

We follow P.-L. Lions’ method, as developed in [40], [43]. First of all, we
replace $u$ with the auxiliary function

$v\coloneqq u+K-C_{0}\left(\tfrac{T-t}{T}\right)\,,\quad\text{where }K=2\lVert
u\rVert_{\infty}+1,\quad C_{0}=2K\,.$

Notice that $\nabla v=\nabla u$, so $v$ solves the same elliptic equation of
$u$, up to an additional term due to the time translation. Moreover, we have
$\lVert v\rVert_{\infty}\leq C(1+\lVert u\rVert_{\infty})$. We set

$z\coloneqq\tfrac{1}{2}\left|\nabla v\right|^{2}+\tfrac{\gamma}{2}v^{2}$

where

$\gamma\coloneqq\frac{\sigma}{(1+\lVert u\rVert_{\infty})^{2}}$

for some small constant $\sigma$ to be chosen later. The goal is to obtain an
upper bound on $z$ by means of the maximum principle. If the maximum occurs at
the boundary, this means that either $t=0$ or $t=T$; here one uses that, by
construction, $v(T)\geq 1$, $v(0)\leq-1$, then reasoning exactly as in [43]
(Thm 3.4, Step 1) one obtains that

$\lVert\nabla v\rVert_{\infty}\leq C(1+\|u\|_{\infty})$

for some
$C=C(\lVert(\varepsilon\log(m_{1})\rVert_{W^{1,\infty}},\lVert(\varepsilon\log(m_{0})\rVert_{W^{1,\infty}},\|\varepsilon\nabla
V\|_{\infty})$, in case of a maximum attained at the extremal times $t=0,T$.

Let us focus on a possibly interior maximum point. From Lemma 4.1, part (i)
(applied to $v$) and part (iii), we have

$\begin{split}&-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}z\right)+\varepsilon\nabla
z\operatorname{\cdot_{g}\\!}\nabla V+2\rho\,z=-\gamma\mathcal{A}(x,\nabla
u)([\partial_{t}v,\nabla v],[\partial_{t}v,\nabla v])]+\\\
&\quad+\left|\nabla\varphi\right|^{2}-|\nabla(\partial_{t}v)|^{2}-\varepsilon\left(\left|\nabla^{2}u\right|^{2}+Ricc_{g}(\nabla
u,\nabla u)+(\nabla^{2}V)(\nabla u,\nabla u)\right)\\\
&\quad+\gamma\rho(K-C_{0}\tfrac{T-t}{T})v\end{split}$ (4.8)

where $\varphi=\frac{1}{2}|\nabla u|^{2}$. By definition of the matrix
$\mathcal{A}(x,\nabla u)$, we get

$\mathcal{A}(x,\nabla u)([\partial_{t}v,\nabla v],[\partial_{t}v,\nabla
v])=\left|-\partial_{t}v+\left|\nabla
v\right|^{2}\right|^{2}+\varepsilon\left|\nabla v\right|^{2}\\\ $

while the definition of $\varphi$ implies

$|\nabla\varphi|^{2}=\left|\nabla z\right|^{2}-2\gamma v\nabla
v\operatorname{\cdot_{g}\\!}\nabla z+\gamma^{2}v^{2}\left|\nabla
v\right|^{2}.$

Inserting the above equalities in (4.8) we get

$\displaystyle-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}z\right)+\varepsilon\nabla
z\operatorname{\cdot_{g}\\!}\nabla V+2\rho
z+\gamma\left|-\partial_{t}v+\left|\nabla
v\right|^{2}\right|^{2}+\gamma\varepsilon\left|\nabla v\right|^{2}$
$\displaystyle\quad=\left|\nabla z\right|^{2}-2\gamma v\nabla
v\operatorname{\cdot_{g}\\!}\nabla z+\gamma^{2}v^{2}\left|\nabla
v\right|^{2}-\left|\nabla(\partial_{t}v)\right|^{2}$
$\displaystyle\qquad-\varepsilon\left(\left|\nabla^{2}u\right|^{2}+Ricc_{g}(\nabla
u,\nabla u)+(\nabla^{2}V)(\nabla u,\nabla
u)\right)+\gamma\rho(K-C_{0}\tfrac{T-t}{T})v$

and since $Ric_{g}$ is bounded below, and $\gamma v^{2}\leq C\sigma$ by the
initial choice, we can estimate the right-hand side obtaining

$\begin{split}&-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}z\right)+\nabla z\operatorname{\cdot_{g}\\!}\nabla
V+2\rho z+\gamma\left|-\partial_{t}v+\left|\nabla
v\right|^{2}\right|^{2}+\gamma\varepsilon\left|\nabla v\right|^{2}\\\
&\quad\leq\left|\nabla z\right|^{2}-2\gamma v\nabla
v\operatorname{\cdot_{g}\\!}\nabla z+C(1+|\nabla v|^{2})\end{split}$ (4.9)

for some constant $C$ depending on $(Ric_{g}+\nabla^{2}V)_{-}$ and on
$\sigma$. Let us focus on the quantity $-\partial_{t}v+|\nabla v|^{2}$: at an
interior maximum point of $z$, we have

$\displaystyle-\partial_{t}v+|\nabla v|^{2}$
$\displaystyle=(\max_{\overline{Q}}z-\partial_{t}u)-\frac{C_{0}}{T}-\frac{\gamma}{2}v^{2}+\frac{1}{2}|\nabla
v|^{2}$

However, thanks to Lemma 4.2, and to the boundary conditions at $t=0,T$, we
have

$\displaystyle\partial_{t}u$
$\displaystyle\leq\sup_{\Sigma_{0}\cup\Sigma_{t}}\partial_{t}u$
$\displaystyle\leq\max_{\overline{Q}}z+\|\delta
u\|_{\infty}+\frac{\gamma}{2}\|v\|_{\infty}^{2}+\|\varepsilon
V\|_{\infty}+\varepsilon\max\left(\|\log(m_{0})\|_{\infty},\|\log(m_{1})\|_{\infty}\right)$

which implies, using $\gamma v^{2}\leq C\sigma$ and (4.5),

$\partial_{t}u-\max_{\overline{Q}}z\leq\overline{K}$

for a certain $\overline{K}>0$ depending only on $\lVert\varepsilon
V\rVert_{\infty},\lVert\varepsilon\log(m_{1})\rVert_{\infty}$ and
$\lVert\varepsilon\log(m_{0})\rVert_{\infty}$. Therefore, we conclude that

$\displaystyle-\partial_{t}v+|\nabla v|^{2}$
$\displaystyle\geq-\overline{K}-\frac{C_{0}}{T}-\frac{\gamma}{2}v^{2}+\frac{1}{2}|\nabla
v|^{2}$ $\displaystyle\geq-K+\frac{1}{2}|\nabla v|^{2}$

where $K=\overline{K}+\frac{C_{0}}{T}+C\sigma$. So, either $|\nabla v|^{2}\leq
4K$ (and then $\max_{\overline{Q}}z\leq 2K+C\frac{\sigma}{2}$) or we have
$-K+\frac{1}{2}|\nabla v|^{2}\geq\frac{1}{4}|\nabla v|^{2}$, and we estimate
$|-\partial_{t}v+|\nabla v|^{2}|^{2}\geq\frac{1}{16}|\nabla v|^{4}$. In this
latter case, looking at (4.9) on a maximum point of $z$, where $\nabla z=0$,
we deduce that

$\frac{\gamma}{16}|\nabla v|^{4}\leq C(1+|\nabla v|^{2})$

which implies $|\nabla v|^{2}\leq C(1+\frac{1}{\gamma})\leq
C(1+\|u\|_{\infty}^{2})$. Thus, we conclude in both cases with an estimate
like (4.7), for the spatial gradient $\nabla u$. Due to Lemma 4.2, this also
yields an estimate for $\partial_{t}u$, and then the full gradient
$\overline{\nabla}u$ is estimated. ∎

###  4.2 Local bound on the density

We next derive a local (in time) version of the gradient estimate. This is not
enough to give a local Lipschitz bound for $u$, but provides with a local
$L^{\infty}$-bound for the density $m$.

######  Proposition 4.4:

Let $u$ be a solution of (4.4). Let $0<a<b<T$ and $\kappa\in(0,\frac{1}{2})$.
There exists a constant $K>0$ (independent of $\rho,\delta$) such that

$\varepsilon\,(\log(m(t))+V)+\kappa|\nabla u|^{2}\leq
K\left(\frac{1}{(t-a)^{2}}+\frac{1}{(b-t)^{2}}\right)\quad\forall t\in(a,b)$
(4.10)

where $K$ depends only on $\kappa,T$, $||u||_{L^{\infty}((a,b)\times M)}$ and
$\|(Ricc_{g}+\nabla^{2}V)_{-}\|_{\infty}$.

###### Proof.

We follow an idea introduced in [43], where a similar result was proved in the
Euclidean case.

Let us consider $z:[a,b]\times M\to\mathbb{R}$ defined by

$z\coloneqq\theta|\nabla
u|^{2}-\partial_{t}u+\gamma\frac{u^{2}}{2},\quad\hbox{where $\theta\in(0,1)$,
$\gamma=\frac{1}{(1+||u||_{L^{\infty}((a,b)\times M)})^{2}}$}.$

By using Lemma 4.1 and denoting $\varphi=\frac{|\nabla u|^{2}}{2}$, we obtain

$\begin{split}-&tr\left(\mathcal{A}(x,\nabla
u)\overline{\nabla}^{2}z\right)+\varepsilon\nabla
z\operatorname{\cdot_{g}\\!}\nabla V+\rho
z\leq-2\theta\,\varepsilon\left(|\nabla^{2}u|^{2}+Ricc_{g}(\nabla u,\nabla
u)+(\nabla^{2}V)(\nabla u,\nabla u)\right)\\\
&+2|\nabla(\partial_{t}u)|^{2}-2\nabla\varphi\operatorname{\cdot_{g}\\!}\nabla(\partial_{t}u)+2\theta\,[|\nabla\varphi|^{2}-|\nabla(\partial_{t}u)|^{2}]-\gamma\,\mathcal{A}(x,\nabla
u)(\overline{\nabla}u,\overline{\nabla}u)\end{split}$ (4.11)

where we notice that

$\displaystyle
2|\nabla(\partial_{t}u)|^{2}-2\nabla\varphi\operatorname{\cdot_{g}\\!}\nabla(\partial_{t}u)+2\theta\,[|\nabla\varphi|^{2}-|\nabla(\partial_{t}u)|^{2}]$
$\displaystyle\quad=-\nabla(\partial_{t}u\operatorname{\cdot_{g}\\!}(2\nabla\varphi-2\nabla(\partial_{t}u))+\theta\,\nabla|\nabla
u|^{2}\operatorname{\cdot_{g}\\!}(2\nabla\varphi-2\nabla(\partial_{t}u))-2\theta|\nabla\varphi-\nabla(\partial_{t}u)|^{2}$
$\displaystyle\quad=\nabla
z\operatorname{\cdot_{g}\\!}(2\nabla\varphi-2\nabla(\partial_{t}u))-2\theta\,|\nabla\varphi-\nabla(\partial_{t}u)|^{2}-\gamma\,u\,\nabla
u\operatorname{\cdot_{g}\\!}(2\nabla\varphi-2\nabla(\partial_{t}u))$
$\displaystyle\quad\leq\frac{1}{\theta}|\nabla
z|^{2}+\frac{1}{\theta}\,\gamma^{2}\,u^{2}\,|\nabla u|^{2}\,.$

By construction of $\mathcal{A}(x,\nabla u)$, we have

$\mathcal{A}(x,\nabla u)(\overline{\nabla}u,\overline{\nabla}u)=||\nabla
u|^{2}-\partial_{t}u|^{2}+\varepsilon|\nabla u|^{2}$

so that, inserting the above estimates in (4.11), we obtain

$\displaystyle-$ $\displaystyle tr\left(\mathcal{A}(x,\nabla
u)\overline{\nabla}^{2}z\right)+\varepsilon\nabla
z\operatorname{\cdot_{g}\\!}\nabla V+\rho
z\leq-2\theta\,\varepsilon\left(|\nabla^{2}u|^{2}+Ricc_{g}(\nabla u,\nabla
u)+(\nabla^{2}V)(\nabla u,\nabla u)\right)$
$\displaystyle\quad+\frac{1}{\theta}|\nabla
z|^{2}+\frac{1}{\theta}\,\gamma^{2}\,u^{2}\,|\nabla u|^{2}-\gamma\,||\nabla
u|^{2}-\partial_{t}u|^{2}-\gamma\,\varepsilon|\nabla u|^{2}\,.$

Using $\gamma\,u^{2}\leq 1$ and the regularity of $V$ we get

$\displaystyle-tr\left(\mathcal{A}(x,\nabla
u)\overline{\nabla}^{2}z\right)+\varepsilon\nabla
z\operatorname{\cdot_{g}\\!}\nabla V+\rho
z\leq\left(\frac{\gamma}{\theta}+\hat{C}\right)|\nabla
u|^{2}+\frac{1}{\theta}|\nabla z|^{2}-\gamma\,||\nabla
u|^{2}-\partial_{t}u|^{2}$ (4.12)

where $\hat{C}$ is a constant depending on the lower bound of
$Ric_{g}+\nabla^{2}V$. Given $L>0$, let

$\psi\coloneqq L\left(\frac{1}{(t-a)^{2}}+\frac{1}{(b-t)^{2}}\right)$

Since $\psi$ blows-up at $t=a,b$, we have that $z-\psi$ admits a maximum point
in $(a,b)\times M$. In such point we have $\nabla z=0$ and

$-tr\left(\mathcal{A}(x,\nabla u)\overline{\nabla}^{2}z\right)\geq-
tr\left(\mathcal{A}(x,\nabla
u)\overline{\nabla}^{2}\psi\right)=-6L\left(\frac{1}{(t-a)^{4}}+\frac{1}{(b-t)^{4}}\right).$

If $L_{0}\coloneqq\max(z-\psi)\geq\frac{1}{2}$, then, at a maximum point of
$z-\psi$ we have

$|\nabla u|^{2}-\partial_{t}u=(1-\theta)|\nabla
u|^{2}+z-\gamma\frac{u^{2}}{2}\geq(1-\theta)|\nabla
u|^{2}+\psi+L_{0}-\frac{1}{2}\geq(1-\theta)|\nabla u|^{2}+\psi>0$

so

$||\nabla u|^{2}-\partial_{t}u|^{2}\geq(1-\theta)^{2}\,|\nabla
u|^{4}+\psi^{2}.$

Using all of these in (4.12), we get

$-6L\left(\frac{1}{(t-a)^{4}}+\frac{1}{(b-t)^{4}}\right)\leq\,\left(\frac{\gamma}{\theta}+\hat{C}\right)|\nabla
u|^{2}-\gamma\,(1-\theta)^{2}|\nabla u|^{4}-\gamma\psi^{2}$

which gives

$\displaystyle\left(\delta\,L^{2}-6L\right)\left(\frac{1}{(t-a)^{4}}+\frac{1}{(b-t)^{4}}\right)$
$\displaystyle\leq\,\left(\frac{\gamma}{\theta}+\hat{C}\right)|\nabla
u|^{2}-\gamma\,(1-\theta)^{2}|\nabla u|^{4}$ $\displaystyle\leq
K(1+\|u\|_{L^{\infty}((a,b)\times M)}^{2})$

for some $K$ only depending on $\theta$ and $\hat{C}$. But the inequality
above cannot hold for any too large $L$ (handling constants with care, this
occurs for $L=O(K[(b-a)\vee 1]^{4}\,(1+\|u\|_{L^{\infty}((a,b)\times
M)}^{2}))$). The conclusion is that we have $\max(z-\psi)\leq\frac{1}{2}$,
hence

$z\leq L\left(\frac{1}{(t-a)^{2}}+\frac{1}{(b-t)^{2}}\right)+\frac{1}{2}\,.$

Using the definition of $z$ leads to

$\theta\,|\nabla u|^{2}-\partial_{t}u\leq
L\left(\frac{1}{(t-a)^{2}}+\frac{1}{(b-t)^{2}}\right)+\frac{1}{2}$

and choosing $\theta>\frac{1}{2}$, from $\partial_{t}u=\frac{1}{2}|\nabla
u|^{2}-\varepsilon(\log(m)+V)$ we obtain (4.10) with
$\kappa=\theta-\frac{1}{2}<\frac{1}{2}$. ∎

###  4.3 Existence of smooth solutions for a penalized problem

We first collect all the above ingredients to show that a penalized version of
the optimality system admits a classical solution. We consider, for
$\delta>0$, the problem

$\begin{cases}-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}u\right)+\varepsilon\nabla
u\operatorname{\cdot_{g}\\!}\nabla V(x)=0&\text{in $Q_{T}$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}=\delta
u+\varepsilon(\log(m_{1})+V(x))&\text{in $\Sigma_{T}$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}+\delta
u=\varepsilon(\log(m_{0})+V(x))&\text{in $\Sigma_{0}$.}\\\ \end{cases}$ (4.13)

which is equivalent, reasoning as in Section 3.2, to the system

$\begin{cases}-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}=\varepsilon(\log(m)+V)&\hbox{in $Q_{T}$}\\\
\partial_{t}m-div_{g}(m\nabla u)=0&\hbox{in $Q_{T}$}\\\ \delta
u(T)=\varepsilon\log(m(T))-\varepsilon\log(m_{1})\,,&\hbox{in $\Sigma_{T}$}\\\
\delta u(0)=\varepsilon\log(m_{0})-\varepsilon\log(m(0))\,&\hbox{in
$\Sigma_{0}$.}\end{cases}$ (4.14)

The auxiliary penalized problem (4.14) has the advantage that the
$L^{\infty}$-norm of $u$ is controlled, for $\delta>0$, see (4.5). The
estimates derived from the elliptic theory (see Appendix A) yield a smooth
solution, with $m>0$, provided the marginals are positive and smooth.

######  Theorem 4.5:

Assume that $V\in W^{2,\infty}(M)$, $m_{0},m_{1}\in C^{1,\alpha}(M)$ and
$m_{0},m_{1}>0$ in $M$. For every $\delta>0$, there exists a unique smooth
solution $(u^{\delta},m^{\delta})$ of (4.14), in the sense that $u\in
C^{2,\alpha}(Q_{T})\cap C^{1,\alpha}(\overline{Q}_{T}),m\in
C^{1,\alpha}(Q_{T})\cap C^{0,\alpha}(\overline{Q}_{T})$, $m>0$ and the
equations are satisfied in a classical sense.

###### Proof.

We first rely on Proposition 7.1 which is proved in the Appendix. This gives a
sequence of solutions $u_{\rho}$ of problem (7.1). By maximum principle, we
have $\|u_{\rho}\|_{\infty}\leq\frac{C}{\delta}$. By the gradient bound proved
in Theorem 4.3, we have that
$\lVert\overline{\nabla}u_{\rho}\rVert_{\infty}\leq C$. By elliptic estimates
(see also Lemma 7.2 below) we deduce first that
$\|u_{\rho}\|_{C^{1,\alpha}(\overline{Q}_{T})}\leq C$, and then, bootstrapping
Schauder estimates, $u_{\rho}$ is bounded in $C^{2,\alpha}$ on any compact
subset of $Q_{T}$. Defining

$m_{\rho}=\exp\left(\frac{-\partial_{t}u_{\rho}+\frac{1}{2}\left|\nabla
u_{\rho}\right|^{2}}{\varepsilon}-V\right)$

we deduce the $C^{1,\alpha}(Q_{T})\cap C^{0,\alpha}(\overline{Q}_{T})$
estimates on $m_{\rho}$ and, in particular, $m_{\rho}$ is uniformly bounded
below due to the gradient bounds for $u_{\rho}$. Passing to the limit as
$\rho\to 0$ gives the desired solution of (4.13), hence of (4.14). ∎

The next step will consist in letting $\delta\to 0$ in (4.14), still assuming
that the marginals $m_{0},m_{1}$ are positive, showing that the minima of
(1.1) are smooth for positive smooth marginals. This step is left to the
stability results of the next Section.

##  5 Existence and regularity of optimal curves

In this section we obtain the existence and the characterization of the minima
of (1.1) in terms of the optimality system (1.2), thus proving Theorem 1.1
stated in the Introduction. We first obtain the existence of smooth minima,
whenever the marginals $m_{0},m_{1}$ are positive and smooth; this is achieved
by passing to the limit as $\delta\to 0$ in problem (4.14) and using the
“elliptic ”Lipschitz estimates of Theorem 4.3. Then we will enlarge the set of
admitted marginals $m_{0}$ and $m_{1}$ to merely $L^{1}$, nonnegative
densities. To this purpose, we will need a relaxed definition of weak solution
to the system (3.1), where merely sub-solutions of the Hamilton-Jacobi
equation are taken into account. This kind of notion of weak solutions,
introduced in [7] (see also [8], [9]) for first order mean-field game systems,
is by now well established also in the context of mean-field transport
problems, see e.g. [20], [41]. In particular, in this latter paper a notion of
trace was developed for functions which are (distributional) sub-solutions of
Hamilton-Jacobi equations, e.g. when $u$ satisfies

$-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}\leq\alpha\,,\qquad\alpha\in L^{1}_{loc}((0,T)\times\Omega)$ (5.1)

in some open set $\Omega$. Relying on the fact that $u$ is nondecreasing in
time, up to an absolutely continuous function, one-sided traces of $u$ were
defined in the sense of limits of measurable functions (with possible range in
$[-\infty,+\infty]$), see [41, Prop 5.6]. In particular, if $\alpha\in
L^{1}(Q_{T})$, any $u$ satisfying (5.1) admits traces at $t=0,t=T$, denoted
below as $u(0),u(T)$ respectively, which are the pointwise limits of
$u(t,\cdot)$ as $t\downarrow 0$ (respectively $t\uparrow T$) in the sense of
measurable functions (for a possibly well defined precise representative of
$u$).

######  Definition 5.1:

A pair $(u,m)$ is a weak solution of

$\begin{cases}-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}=\varepsilon(\log(m)+V(x))\quad&\text{in $Q_{T}$}\\\
\partial_{t}m-div_{g}(m\nabla u)=0&\text{in $Q_{T}$}\\\ m(0,\cdot)=m_{0},\quad
m(T,\cdot)=m_{1}&\text{in $M$}\end{cases}$ (5.2)

if $m\in C^{0}([0,T];\mathcal{P}(M))\cap L^{1}(Q_{T})$ with $m(0)=m_{0},$
$m(T)=m_{1}$ and $\log(m)\in L^{1}_{loc}((0,T);L^{1}(M))$;
$u\in L^{2}_{loc}((0,T);H^{1}(M))$ and in addition $m\left|\nabla
u\right|^{2}\in L^{1}(Q_{T})$, $m\log m\in L^{1}(Q_{T})$ and $(u,m)$ satisfy

  1. i)

$u$ is a weak sub-solution satisfying, in the sense of distributions,

$-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}\leq\varepsilon(\log(m)+V(x))\quad\text{in $Q_{T}$}$

  2. ii)

$m$ is a weak solution satisfying, in the sense of distributions, the
continuity equation

$\partial_{t}m-div_{g}(m\,\nabla u)=0\quad\text{in $Q_{T}$}$

  3. iii)

$(u,m)$ satisfy the identity

$\int_{M}m_{0}u(0)dx-\int_{M}m_{1}u(T)dx=\int_{0}^{T}\\!\\!\\!\\!\int_{M}\left(\frac{1}{2}\left|\nabla
u\right|^{2}m+\varepsilon m(\log m+V)\right)\,dxdt$ (5.3)

where $u(0),u(T)$ are the one-sided traces of $u$ mentioned above.

The existence of weak solutions to (5.2) will be produced through a stability
argument, which relies on several steps. Some crucial a priori estimates are
given in the following lemma, which applies to solutions of both (4.14) and
(5.2) (corresponding to $\delta=0$).

######  Lemma 5.2:

Assume that $(u,m)$ is a smooth solution to (4.14), for some $\delta\in[0,1]$,
and define $\hat{u}:=u-\int_{M}u(T)m_{1}$. Let $\lambda$ be given by (2.2).
There exists a constant $\hat{C}=\hat{C}(M,d,\lambda,T,\|V\|_{\infty})$ such
that

$\displaystyle-\frac{\hat{C}}{t}\leq\hat{u}(t,x)\qquad\forall(t,x)\in(0,T]\times
M\,,$ (5.4)
$\displaystyle\hat{u}(t,x)\leq\frac{\hat{C}}{T-t}\qquad\forall(t,x)\in[0,T)\times
M$

and

$\begin{split}&\int_{M}\hat{u}(0)m_{0}\,dx\geq-\varepsilon\log\left(\int_{M}e^{-V}dx\right)\,\\\
&\int_{0}^{T}\\!\\!\\!\\!\int_{M}\left(m\,|\nabla u|^{2}+\varepsilon m\log
m\right)dxdt\leq\hat{C}\,.\end{split}$ (5.5)

In addition, for any $0<a<b<T$ there exists $C$, depending on $\hat{C}$ and
additionally on $a,(T-b)$, such that

$\int_{a}^{b}\\!\\!\\!\int_{M}|\nabla
u|^{2}\,dxdt+\varepsilon\int_{a}^{b}\\!\\!\\!\int_{M}|\log m|\,dxdt\leq C\,.$
(5.6)

Finally, if $m_{0},m_{1}>0$ on $M$, then there exists $K$, depending on
$\hat{C}$ and additionally on $\min\limits_{M}[\varepsilon\log(m_{0}e^{V})]$
and $\min\limits_{M}[\varepsilon\log(m_{1}e^{V})]$, such that

$||\hat{u}||_{L^{\infty}([0,T]\times M)}<K\,.$ (5.7)

###### Proof.

By definition of $\hat{u}$, we have

$\displaystyle\int_{0}^{T}\\!\\!\int_{M}\left(\frac{1}{2}\left|\nabla
u\right|^{2}m+\varepsilon m(\log
m+V)\right)\,dxdt=\int_{M}m(0)u(0)dx-\int_{M}m(T)u(T)dx$
$\displaystyle\qquad=\int_{M}(m(0)-m_{0})u(0)dx-\int_{M}(m(T)-m_{1})u(T)dx+\int_{M}m_{0}\hat{u}(0)\,dx$

This implies, using the initial-terminal conditions of (4.14),

$\begin{split}&\int_{M}m_{0}\hat{u}(0)dx=\int_{0}^{T}\\!\\!\int_{M}\left(\frac{1}{2}\left|\nabla
u\right|^{2}m+\varepsilon m(\log m+V)\right)\,dxdt\\\
&\quad+\frac{\varepsilon}{\delta}\int_{M}(m_{0}-m(0))[\log(m_{0})-\log(m(0))]\,dx\\\
&\quad+\frac{\varepsilon}{\delta}\int_{M}(m(T)-m_{1})[\log
m(T)-\log(m_{1})]\,dx\,,\end{split}$ (5.8)

where last two terms should be treated as zero in case that $\delta=0$. In
particular, since the right-side is bounded below, we deduce that

$\begin{split}\int_{M}m(0)\hat{u}(0)dx&\geq\int_{0}^{T}\\!\\!\int_{M}\left(\frac{1}{2}\left|\nabla
u\right|^{2}m+\varepsilon m(\log m+V)\right)\,dxdt\\\
&\geq-\varepsilon\,\log\left(\int_{M}e^{-V}dx\right)\end{split}$ (5.9)

where we used Jensen’s inequality in the last step. Of course, $\hat{u}$
satisfies the same Hamilton-Jacobi equation as $u$, from which we derive now
all the estimates.

At first, for any fixed $x_{0}\in M$, letting $\delta_{x_{0}}$ be the Dirac’s
delta concentrated on $x_{0}$, we consider the solution $\mu$ to the
continuity equation

$\begin{cases}\partial_{t}\mu-div_{g}(\mu v)=0&\text{in $(s,T)\times M$}\\\
\mu(T)=m_{1},\mu(s)=\delta_{x_{0}}&\text{in $M$}\\\ \end{cases}$

which was built in Proposition 2.2. Such a curve satisfies the estimate

$\int^{T}_{s}\\!\\!\\!\int_{M}\left(\frac{\left|v\right|^{2}}{2}\,\mu+\varepsilon\mu(\log\mu+V)\right)dxdt\leq\frac{K}{T-s}$

with $K$ depending only on $M,d,\lambda,T$ and $\|V\|_{\infty}$, but
independent of $x_{0}$. We multiply by $\mu$ the equation of $\hat{u}$ and we
integrate over $(s,T)\times M$: we get

$\displaystyle\hat{u}(x_{0})$
$\displaystyle=\int^{T}_{s}\\!\\!\\!\int_{M}v\cdot\nabla
u\,\mu-\int^{T}_{s}\\!\\!\\!\int_{M}\frac{1}{2}\left|\nabla
u\right|^{2}\,\mu+\varepsilon\int^{T}_{s}\\!\\!\\!\int_{M}(\log(m)+V)\mu\,dxdt$
$\displaystyle\leq\int^{T}_{s}\\!\\!\\!\int_{M}\frac{1}{2}\left|v\right|^{2}\,\mu+\varepsilon\int^{T}_{s}\\!\\!\\!\int_{M}(\log(m)+V)\mu\,dxdt\,.$

Using that $a\log(b)\leq a\log(a)+b$ for every $a,b\in(0,+\infty)$ we obtain

$\displaystyle\hat{u}(x_{0})$
$\displaystyle\leq\int^{T}_{s}\\!\\!\\!\int_{M}\frac{1}{2}\left|v\right|^{2}\,\mu+\varepsilon\int^{T}_{s}\\!\\!\\!\int_{M}\mu(\log(\mu)+V)\,dxdt+\varepsilon(T-s)$
$\displaystyle\leq\frac{K}{T-s}+\varepsilon(T-s)\,.$

So there exists a constant $\hat{C}>0$, depending on $T$, $\|V\|_{\infty}$ and
$M$, but independent of $\varepsilon$ and $x_{0}$, such that for every
$t\in[0,T)$

$\hat{u}(t,x_{0})\leq\frac{\hat{C}}{T-t}\,.$ (5.10)

Reasoning in a similar way, namely using an analogous curve between $m_{0}$
and $\delta_{x_{0}}$, and the bound (5.9), we conclude that there exists a
constant $\hat{C}>0$ such that for every $t\in(0,T]$

$-\frac{\hat{C}}{t}\leq\hat{u}(t,x_{0}).$ (5.11)

By the arbitrariness in the choice of $x_{0}\in M$, we get (5.4).

Fix now $0<a<b<T$, from the equation of $\hat{u}$ we have

$\displaystyle\int_{M}\hat{u}(a,\cdot)\,dx-\int_{M}\hat{u}(b,\cdot)\,dx$
$\displaystyle+\int_{a}^{b}\\!\\!\int_{M}\frac{1}{2}\left|\nabla
u\right|^{2}\,dxdt$
$\displaystyle\leq-\varepsilon\int_{a}^{b}\\!\\!\int_{M}|\log(m)|dxdt+\varepsilon\,c_{M}(1+\|V\|_{\infty})\,.$

Using (5.10) and (5.11), we obtain estimate (5.6). Coming back to (5.9) and
using the upper bound (5.10), we also get (5.5).

Finally, to get (5.7) we use the comparison principle on the equation (4.13).
In fact, the linear function $\psi:=\frac{\hat{C}}{T}+Bt$ is clearly a
solution of the same equation, and it is a strict supersolution at $t=T$ if
$\varepsilon(\log(m_{1}e^{V}))>-B$. This is possible if $m_{1}>0$, choosing
$B$ sufficiently large. Since $\psi(0)>\hat{u}(0)$ by (5.10), one can compare
$\hat{u}$ with $\psi$ (note that $\hat{u}$ satisfies the same condition as $u$
at $t=T$ up to a bounded term, and $B$ can be chosen possibly larger to
compensate). Hence $\hat{u}\leq\frac{\hat{C}}{T}+Bt$. Similarly we reason to
get the estimate from below using the positivity of $m_{0}$, and we conclude
with the global bound (5.7). ∎

We notice that the local bounds of $\hat{u}$, given by (5.4), are totally
independent of $m$. This allows us to infer the local boundedness of $m$ too,
as a consequence of Proposition 4.4.

######  Corollary 5.3:

Under the assumptions of Lemma 5.2, given any $0<a<b<T$ there exists a
constant
$C=C(M,d,\lambda,T,\|V\|_{\infty},\|(Ric_{g}+D^{2}V)_{-}\|_{\infty},\varepsilon,a,T-b)$
such that

$\|m(t)\|_{L^{\infty}((a,b)\times M)}\leq C\,.$

###### Proof.

By estimates (5.4), we have $\|\hat{u}\|_{L^{\infty}((a,b)\times
M)}\leq\hat{C}\,\max(a^{-1},(T-b)^{-1})$. Hence, from (4.10) we deduce that
$\varepsilon\,\log(m(t))$ is bounded above for $t\in(a,b)$ by some constant
depending on $\hat{C},T,\max(a^{-1},(T-b)^{-1})$ and
$\|(Ric_{g}+D^{2}V)_{-}\|_{\infty}$, which yields the conclusion. ∎

Thanks to the global bound (5.7), we can now pass to the limit as $\delta\to
0$ in (4.14), obtaining the existence of smooth minimizers for positive smooth
marginals.

######  Theorem 5.4:

Assume that $m_{0},m_{1}\in W^{1,\infty}(M)$ and $m_{0},m_{1}>0$ in $M$. Let
$V\in W^{2,\infty}(M)$, $\varepsilon>0$. Then there exists a (unique) smooth
solution $(u,m)$ of the system (1.2) such that $\int_{M}u(T)m_{1}=0$, $u\in
C^{2,\alpha}(Q_{T})\cap C^{1,\alpha}(\overline{Q}_{T})$, $m\in
C^{1,\alpha}(Q_{T})\cap C^{0,\alpha}(\overline{Q}_{T})$, $\alpha\in(0,1)$. In
addition we have $m>0$ in $\overline{Q}_{T}$, $u$ is a classical solution of
the elliptic equation (3.17), and $(m,\nabla u)$ is the unique minimum of the
functional $\mathcal{F}_{\varepsilon}$ in (1.1).

Finally, if $V\in C^{k,\alpha}(M)$, we have $u\in C^{k+1,\alpha}(M),m\in
C^{k,\alpha}(M)$.

###### Proof.

In a first step, we take $m_{0},m_{1}\in C^{1,\alpha}(M)$ and positive on $M$.
By Theorem 4.5, problem (4.14) admits a smooth solution
$(u^{\delta},m^{\delta})$, and we set as before
$\hat{u}^{\delta}=u^{\delta}-\int_{M}u^{\delta}(T)\,m_{1}$. By (5.7), we have
that $\hat{u}^{\delta}$ is uniformly bounded in $Q_{T}$. Then, by Theorem 4.3,
we deduce that $\hat{u}^{\delta}$ is uniformly bounded in Lipschitz norm
(time-space). By elliptic estimates (same as in Lemma 7.2), we have that
$\hat{u}^{\delta}$ is bounded in $C^{1,\alpha}(\overline{Q}_{T})$ and the
bound only depends on
$\|\log(m_{0})\|_{W^{1,\infty}(M)},\|\log(m_{1})\|_{W^{1,\infty}(M)}$ (and of
course on $\varepsilon,\|V\|_{W^{2,\infty}}$). We also have interior local
bounds on $\hat{u}^{\delta}$ in $C^{2,\alpha}$, because the elliptic equation
have coefficients bounded in $C^{0,\alpha}$. Therefore, by compactness, we
deduce that $\hat{u}^{\delta}$ converges to some $u\in C^{2,\alpha}(Q_{T})\cap
C^{1,\alpha}(\overline{Q}_{T})$, which is a classical solution of the elliptic
equation (3.17). At the boundary, e.g. at $t=T$, we have

$\begin{split}-\partial_{t}\hat{u}^{\delta}&+\frac{1}{2}\left|\nabla\hat{u}^{\delta}\right|^{2}=\varepsilon(\log(m_{1})+V(x))+\delta\hat{u}^{\delta}+\delta\int_{M}u^{\delta}(T)m_{1}\\\
&=\varepsilon(\log(m_{1})+V(x))+\delta\hat{u}^{\delta}+\varepsilon\int_{M}(\log(m^{\delta}(T))-\log(m_{1}))m_{1}\,.\end{split}$
(5.12)

However, by (5.8) and the bound on $\hat{u}^{\delta}$, we know that

$\int_{M}(\log(m^{\delta}(T))-\log(m_{1}))(m^{\delta}(T)-m_{1})\leq
C\,\delta\,\,\mathop{\to}^{\delta\to 0}0$

which implies that $m^{\delta}(T)\to m_{1}$. In particular, last term in
(5.12) vanishes as $\delta\to 0$, and since we also have
$\delta\hat{u}^{\delta}\to 0$ we conclude that $u$ satisfies the boundary
condition at $t=T$

$-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}=\varepsilon(\log(m_{1})+V(x))\,.$

We argue in the same way for $t=0$ and we conclude that the limit $u$
satisfies the elliptic problem (4.4) with $\rho=0,\delta=0$. Furthermore, we
notice that $u$ satisfies an elliptic equation where the second order
coefficients only depend on $\nabla u$, and the first order coefficients
depend on $\nabla V$. Hence, by the interior Schauder regularity and a
boostrap argument, we have $u\in C^{k+1,\alpha}(M)$ provided $V\in
C^{k,\alpha}(M)$, $k\geq 1$.

Finally, defining

$m:=e^{-V(x)}\,\exp\left(\frac{-\partial_{t}u+\frac{1}{2}|\nabla
u|^{2}}{\varepsilon}\right)\,,$

we have $m\in C^{1,\alpha}(Q_{T})\cap C^{0,\alpha}(\overline{Q}_{T})$, $m>0$
and $m(0)=m_{0},m(T)=m_{1}$. In other words, $(u,m)$ is a smooth solution of
system (1.2) (unique, with the normalization of $u$), and is the minimum of
the functional $\mathcal{F}_{\varepsilon}$. This is also the unique minimum,
as we will prove in more generality in our next results. As a last remark, the
result extends by approximation to positive marginals $m_{0},m_{1}\in
W^{1,\infty}(M)$ since in fact all the estimates above remain true. ∎

We will now extend the existence result to the case of general $L^{1}$
marginals. We will need some more a priori estimates and new compactness
arguments. To this purpose, we first make use of the displacement convexity
estimates of Proposition 3.3 to derive a local $L^{2}$-bound on $\nabla m$.

######  Lemma 5.5:

Let $(u,m)$ be a smooth solution of the system (3.1). For every $0<a<b<T$
there exist a constant $C=C(M,d,\lambda,T,b-a,\lVert V\rVert_{W^{1,\infty}})$
such that

$\displaystyle\int_{a}^{b}\\!\\!\int_{M}\left|\nabla\sqrt{m}\right|^{2}\,dxdt$
$\displaystyle\leq C\,\frac{|\log\varepsilon|}{\varepsilon}\,.$

###### Proof.

First, we recall inequality (3.11) which implies

$\displaystyle\frac{d^{2}}{dt^{2}}\int_{M}m\log m$
$\displaystyle\geq-2\lambda\varepsilon\int_{M}m\log
mdx+\frac{\varepsilon}{2}\int_{M}\frac{1}{m}\left|\nabla m\right|^{2}\,dx-C$

for some $C$ depending on $M,d,\lambda,T,\|V\|_{W^{1,\infty}}$, and
independent of $\varepsilon\leq 1$. Now we fix $t_{0}\in(0,T)$, and
$R<R_{0}\coloneqq\min(t_{0},T-t_{0})$; then, for $\tau\in(0,R)$ we let
$\xi(t)$ be a smooth cut-off function such that

$\begin{cases}\xi(t)=1&\text{if }t\in(t_{0}-\tau,t_{0}+\tau)\\\
\xi(t)=0&\text{if }\left|t-t_{0}\right|>R\\\
\left|\xi^{\prime}(t)\right|^{2}+\left|\xi^{\prime\prime}(t)\right|\leq\alpha_{\xi}\end{cases}$

for a certain $\alpha_{\xi}>0$. Then we have

$\displaystyle\frac{d^{2}}{dt^{2}}\left(\xi^{2}\int_{M}m\log m\,dx\right)\geq$
$\displaystyle-2\lambda\varepsilon\xi^{2}\int_{M}m\log
m\,dx+\frac{\varepsilon}{2}\int_{M}\xi^{2}\frac{\left|\nabla
m\right|^{2}}{m}\,dx-C\,\xi^{2}$
$\displaystyle+4\xi\xi^{\prime}\frac{d}{dt}\int_{M}m\log m\,dx+2(\xi^{\prime
2}+\xi\xi^{\prime\prime})\int_{M}m\log m\,dx\,.$

If we integrate in $(0,T)$ both sides we get

$\displaystyle\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}\xi^{2}\left|\nabla\sqrt{m}\right|^{2}\,dxdt\leq\,$
$\displaystyle C_{t_{0},T}\left[\int_{0}^{T}\\!\\!\\!\\!\int_{M}m\log
m\,dxdt+C\right],$

for a possibly different constant $C$. Since
$\int_{0}^{T}\\!\\!\\!\\!\int_{M}m\log(m)$ is estimated by (3.9), we conclude.
∎

We will also use the following stability result.

######  Lemma 5.6:

Let $(u^{n},m^{n})$ be a sequence of smooth solutions of (3.1), possibly for
different parameters $\varepsilon_{n}\to\varepsilon\geq 0$. Assume that
$u_{n}$ satisfies (5.4) for some constant $\hat{C}$ independent of $n$. Let
$(u,m)$ be such that $u^{n}\to u$ weakly in $L^{2}((a,b);H^{1}(M))$, for any
$0<a<b<T$, and $m^{n}\to m$ weakly in $L^{1}(Q_{T})$. Then we have:

  1. 1.

$u$ satisfies, in the sense of distributions,

$-\partial_{t}u+\frac{1}{2}|\nabla u|^{2}\leq\varepsilon(\log
m+V)\qquad\hbox{in $(0,T)\times M$.}$ (5.13)

  2. 2.

For every sequences $m_{0n},m_{1n}$ such that $m_{0n}\to m_{0}$, $m_{1n}\to
m_{1}$ strongly in $L^{1}(M)$, we have

$\limsup_{n\to\infty}\int_{M}u^{n}(0)m_{0n}\leq\int_{M}u(0)dm_{0}\,,$
$\liminf_{n\to\infty}\int_{M}u^{n}(T)m_{1n}\geq\int_{M}u(T)dm_{1}\,.$

  3. 3.

For every $(\mu,v)$ which solves (2.3) and such that $\mu(t)\in L^{1}(M)$ for
every $t$ and $\mu\log\mu\in L^{1}(Q_{T})$, it holds

$\begin{split}\int_{M}u(s)\mu(s)\,dx-\int_{M}u(t)\,\mu(t)\,dx&\leq\int_{s}^{t}\\!\\!\\!\int_{M}[\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla
u-\frac{1}{2}|\nabla u|^{2}\,\mu]\,dxd\tau\\\
&\quad+\varepsilon\int_{s}^{t}\\!\\!\\!\int_{M}\left(\log m+V\right)\mu
dxd\tau\end{split}$ (5.14)

for every $0\leq s<t\leq T$.

######  Remark 5.7:

We notice that (5.13) implies

$-\partial_{t}u+\frac{1}{2}|\nabla u|^{2}\leq\varepsilon(m+V)\,\,\,\in
L^{1}(Q_{T})$

hence $u$ satisfies (5.1) for some $\alpha\in L^{1}(Q_{T})$ and admits one-
sided traces up to $t=0$ and $t=T$ (those traces are used in items 2,3 above).

###### Proof.

By definition, $u^{n}$ satisfies

$\int_{0}^{T}\\!\\!\\!\\!\int_{M}u_{n}\,\partial_{t}\varphi\,dxdt+\frac{1}{2}\int_{0}^{T}\\!\\!\\!\\!\int_{M}|\nabla
u^{n}|^{2}\,\varphi\,dxdt=\int_{0}^{T}\\!\\!\\!\\!\int_{M}\varepsilon(\log(m^{n})+V)\varphi\,dxdt$

for every $\varphi\in C^{1,0}_{c}(Q_{T})$. In particular the integrals are
restricted to some interval $(a,b)$ containing the support of $\varphi$, with
$0<a<b<T$. Then, by simply using the weak convergence of $\nabla u_{n}$ in
$L^{2}((a,b)\times M)$ and $m^{n}$ in $L^{1}((a,b)\times M)$, and the weak
lower semicontinuity for the convex functions $|p|^{2}$ and $-\log(m)$,
respectively, we conclude that $u$ satisfies (5.13), in distributional sense.

To observe the relaxation on the initial traces, we fix $k\in\mathbb{N}$ and
consider the truncations $u_{n,k}\coloneqq\max\\{u_{n},-k\\}$. By a standard
argument in Sobolev spaces (see e.g. Lemma 5.3 in [41]), we have that
$u_{n,k}:=\max\\{u_{n},-k\\}$ are Lipschitz functions that satisfy

$\begin{split}-\partial_{t}u_{n,k}+\frac{1}{2}|\nabla
u_{n,k}|^{2}&\leq\mathds{1}_{\\{u_{n}\geq-k\\}}\varepsilon_{n}\left(\log
m_{n}+V\right)\\\ &\leq\varepsilon_{n}\,(m_{n}+|V|)\,.\end{split}$ (5.15)

For every $k\in\mathbb{N}$, let $\xi_{k}$ be the piecewise linear function
such that

$\begin{cases}\xi_{k}(s)=1&\forall s\in\left[0,\frac{\hat{C}}{k}\right]\\\
\xi_{k}(s)=0&\forall s\in\left[2\frac{\hat{C}}{k},T\right]\\\
\xi_{\delta,k}^{\prime}(s)=-\frac{k}{\hat{C}}&\forall
s\in\left[\frac{\hat{C}}{k},2\frac{\hat{C}}{k}\right]\\\ \end{cases}$

where $\hat{C}$ is the same constant which appears in (5.4). We now fix a
positive $\varphi\in L^{\infty}(M)$ and we multiply (5.15) by
$\phi=\xi_{k}\varphi$. Integrating by parts the time derivative we get

$\displaystyle\int_{M}u_{n,k}(0,\cdot)\varphi
dx-\frac{k}{\hat{C}}\int_{\frac{\hat{C}}{k}}^{2\frac{\hat{C}}{k}}\\!\\!\\!\\!\int_{M}u_{n,k}\varphi\,dxdt$
$\displaystyle\leq C\,\frac{\|\varphi\|_{\infty}}{k}\,.$

The sequence $u_{n,k}(0,\cdot)$ is uniformly bounded in $n$ by construction,
let $\chi_{k}$ be its weak-* limit, up to subsequences, in $L^{\infty}(M)$. By
(5.4), we note that $u_{n,k}=u_{n}$ for $t\geq\frac{\hat{C}}{k}$. Then we pass
to the limit for $n\rightarrow+\infty$, using the weak convergence of $u_{n}$,
and we get

$\int_{M}\chi_{k}\,\varphi
dx-\frac{k}{\hat{C}}\int_{\frac{\hat{C}}{k}}^{2\frac{\hat{C}}{k}}\\!\\!\\!\\!\int_{M}u\,\varphi\,dxdt\leq
C\,\frac{\|\varphi\|_{\infty}}{k}.$ (5.16)

From equation (5.13), setting
$F(t,x):=\varepsilon\int_{0}^{t}(m(s,x)+V(x))\,ds$, we know that $u+F$ is
nondecreasing in time. Hence

$\frac{k}{\hat{C}}\int_{\frac{\hat{C}}{k}}^{2\frac{\hat{C}}{k}}\\!\\!\\!\\!\int_{M}u\,\varphi\,dxdt\leq\int_{M}u(2\frac{\hat{C}}{k},\cdot)\varphi\,dx-\frac{k}{\hat{C}}\int_{\frac{\hat{C}}{k}}^{2\frac{\hat{C}}{k}}\\!\\!\\!\\!\int_{M}\left(F(t,x)-F(2\frac{\hat{C}}{k},x)\right)\varphi\,dxdt$

and last term vanishes as $k\to\infty$ because $F$ is a primitive of a $L^{1}$
function. Therefore, we have

$\limsup_{k\to\infty}\frac{k}{\hat{C}}\int_{\frac{\hat{C}}{k}}^{2\frac{\hat{C}}{k}}\\!\\!\\!\\!\int_{M}u\,\varphi\,dxdt\leq\limsup_{k\to\infty}\int_{M}u(2\frac{\hat{C}}{k},\cdot)\varphi\,dx\leq\int_{M}u(0)\,\varphi\,dx$

where we used the pointwise convergence of $u(t,\cdot)$ as $t\to 0^{+}$ and
the fact that $u$ is bounded above by (5.4), which allows us to apply Fatou’s
lemma. Finally, letting $k\to\infty$ in (5.16), we obtain

$\limsup_{k\to\infty}\int_{M}\chi_{k}\varphi dx\leq\int_{M}u(0)\,\varphi\,dx$
(5.17)

for every $\varphi\in L^{\infty}(M)$. Let us define now $T_{j}(f)=\min(f,j)$
the truncation operator in $L^{1}(M)$. We recall that $m_{0,n}$ converges
strongly to $m_{0}$ in $L^{1}(M)$ and $u(0)$ is bounded above, so

$\displaystyle\limsup_{n}\int_{M}m_{0,n}u_{n}(0)dx$
$\displaystyle\leq\limsup_{n}\int_{M}T_{j}(m_{0,n})u_{n}(0)dx$
$\displaystyle\quad+\limsup_{n}\int_{M}(m_{0,n}-T_{j}(m_{0,n}))u_{n}(0)dx$
$\displaystyle\leq\limsup_{n}\int_{M}T_{j}(m_{0,n})u_{n,k}(0)dx$
$\displaystyle\quad+\frac{\hat{C}}{T}\limsup_{n}\int_{M}(m_{0,n}-T_{j}(m_{0,n}))dx$
$\displaystyle\leq\int_{M}T_{j}(m_{0})\chi_{k}dx+\frac{\hat{C}}{T}\int_{M}(m_{0}-T_{j}(m_{0}))dx\,.$

Using (5.17) with $\varphi=T_{j}(m_{0})$ we obtain, letting $k\to\infty$,

$\limsup_{n}\int_{M}m_{0,n}u_{n}(0)dx\leq\int_{M}u(0)\,T_{j}(m_{0})\,dx+\frac{\hat{C}}{T}\int_{M}(m_{0}-T_{j}(m_{0}))dx\,.$

Letting finally $j\to\infty$, we get

$\limsup_{n}\int_{M}m_{0,n}u_{n}(0)dx\leq\int_{M}u(0)\,m_{0}\,dx\,.$

Similarly we argue for $t=T$, using now $u_{n,k}=\min(u_{n},k)$.

We are left to prove (5.14). To this purpose, for any $f\in L^{1}(Q_{T})$, we
denote by $f_{h},f_{-h}$ the time-average functions
$f_{h}:=\frac{1}{h}\int_{t}^{t+h}f(x,s)ds$ and
$f_{-h}:=\frac{1}{h}\int_{t-h}^{t}f(x,s)ds$. Integrating (5.15) in $(t,t+h)$
and dividing by $h$, we obtain, by means of Jensen inequality,

$-\partial_{t}(u_{n,k})_{h}+\frac{1}{2}|\nabla(u_{n,k})_{h}|^{2}\leq\frac{1}{h}\int_{t}^{t+h}\left\\{\mathds{1}_{\\{u_{n}\geq-k\\}}\varepsilon_{n}\left(\log
m_{n}+V\right)\right\\}ds\,.$

Let $(\mu,v)$ be any solution of the continuity equation such that $\mu(t)\in
L^{1}(M)$ for every $t$, and $\mu\log(\mu)\in L^{1}(Q_{T})$. By a density
argument, any Lipschitz function $\varphi$ can be used as test function in the
continuity equation; hence, multiplying by $u_{n,k}$, we get (for any $0\leq
s<r<T-h$)

$\begin{split}&\int_{M}(u_{n,k})_{h}(s)\mu(s)\,dx-\int_{M}(u_{n,k})_{h}(r)\,\mu(r)\,dx\\\
&\quad\qquad\leq\int_{s}^{r}\\!\\!\\!\int_{M}\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla(u_{n,k})_{h}-\frac{1}{2}|\nabla(u_{n,k})_{h}|^{2}\,\mu\,dxd\tau\\\
&\quad\qquad\quad+\varepsilon_{n}\int_{s}^{r}\\!\\!\\!\int_{M}\frac{1}{h}\int_{\tau}^{\tau+h}\left\\{\mathds{1}_{\\{u_{n}\geq-k\\}}\left(\log
m_{n}+V\right)\right\\}\mu(\tau)dsdxd\tau\,.\end{split}$ (5.18)

Now we let $n\to\infty$. Recall that $u_{n}$ is bounded in
$L^{2}((a,b);H^{1}(M))$, and, using (5.6), we have in fact $\partial_{t}u_{n}$
bounded in $L^{1}((a,b);L^{1}(M))$; using classical compactness results (see
[45]), we deduce that $u_{n}$ is compact in $L^{2}((a,b);L^{2}(M))$. Moreover,
$u_{n}$ is locally uniformly bounded (and bounded above up to $t=0$); since
$\mu(t)\in L^{1}(M)$, we deduce that

$\int_{M}(u_{n,k})_{h}(t)\,\mu(t)\,dx\mathop{\to}^{n\to\infty}\int_{M}(u_{k})_{h}(t)\,\mu(t)\,dx$

for any $t\in[0,T)$, where $u_{k}=\max(u,-k)$. Moreover, by weak convergence
of $u_{n}$ in $L^{2}((a,b);H^{1}(M))$, for any small $\eta>0$ we have

$\displaystyle\limsup_{n\to\infty}$
$\displaystyle\int_{s}^{r}\\!\\!\\!\int_{M}\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla(u_{n,k})_{h}-\frac{1}{2}|\nabla(u_{n,k})_{h}|^{2}\,\mu\,dxd\tau$
$\displaystyle\leq\limsup_{n\to\infty}\int_{s+\eta}^{r}\\!\\!\int_{M}\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla(u_{n,k})_{h}-\frac{1}{2}|\nabla(u_{n,k})_{h}|^{2}\,\mu\,dxd\tau+\int_{s}^{s+\eta}\\!\\!\\!\int_{M}\frac{1}{2}\mu|v|^{2}\,dxd\tau$
$\displaystyle\leq\int_{s+\eta}^{r}\\!\\!\int_{M}\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla(u_{k})_{h}-\frac{1}{2}|\nabla(u_{k})_{h}|^{2}\,\mu\,dxd\tau+\int_{s}^{s+\eta}\\!\\!\\!\int_{M}\frac{1}{2}\mu|v|^{2}\,dxd\tau\,,$

and letting $\eta\to 0$ Fatou’s lemma yields

$\displaystyle\limsup_{n\to\infty}$
$\displaystyle\int_{s}^{r}\\!\\!\\!\int_{M}\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla(u_{n,k})_{h}-\frac{1}{2}|\nabla(u_{n,k})_{h}|^{2}\,\mu\,dxd\tau$
$\displaystyle\qquad\qquad\leq\int_{s}^{r}\\!\\!\\!\int_{M}\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla(u_{k})_{h}-\frac{1}{2}|\nabla(u_{k})_{h}|^{2}\,\mu\,dxd\tau\,.$

We are left with the last integral in (5.18). Of course, if
$\varepsilon_{n}\to 0$, using $\mu\log m_{n}\leq\mu\log\mu+m_{n}$, we estimate

$\displaystyle\varepsilon_{n}\int_{s}^{r}\\!\\!\\!\int_{M}\frac{1}{h}\int_{\tau}^{\tau+h}\left\\{\mathds{1}_{\\{u_{n}\geq-k\\}}\left(\log
m_{n}+V\right)\right\\}\mu(\tau)dsdxd\tau$
$\displaystyle\qquad\leq\varepsilon_{n}\int_{s}^{r}\\!\\!\\!\int_{M}\frac{1}{h}\int_{\tau}^{\tau+h}\left(m_{n}+\mu\log\mu+V\,\mu\right)dsdxd\tau\mathop{\to}^{n\to\infty}0\,.$

So we suppose that $\varepsilon_{n}\to\varepsilon>0$. In this case, from Lemma
5.5 we get that $\sqrt{m_{n}}$ is bounded in $L^{2}((a,b);H^{1}(M))$, for any
$0<a<b<T$. Since $m_{n}|\nabla u_{n}|^{2}$ is bounded in $L^{1}(Q_{T})$ and

$(\sqrt{m_{n}})_{t}=\frac{1}{2}div_{g}(\sqrt{m_{n}}\nabla
u_{n})+\frac{1}{2}\nabla u_{n}\operatorname{\cdot_{g}\\!}\nabla\sqrt{m_{n}}$

we also deduce that $(\sqrt{m_{n}})_{t}$ is bounded in
$L^{2}((a,b);(H^{1}(M))^{*})+L^{1}((a,b)\times M)$. By classical compactness
results (see [45]), we infer that $\sqrt{m_{n}}$ is strongly compact in
$L^{2}((a,b)\times M)$, which means that $m_{n}$ is relatively compact in the
strong $L^{1}$-topology, locally in time, and converges almost everywhere, up
to subsequences. In particular, still using that $\mu\log
m_{n}\leq\mu\log\mu+m_{n}$, we can apply Fatou’s lemma in the last integral in
(5.18) (notice that $\mathds{1}_{\\{u_{n}\geq-k\\}}$ converges to
$\mathds{1}_{\\{u\geq-k\\}}$ for almost every $k$, which we can suppose to be
the case). Finally, by (5.18) we obtain, letting $n\to\infty$:

$\displaystyle\int_{M}(u_{k})_{h}(s)\mu(s)\,dx-\int_{M}(u_{k})_{h}(r)\,\mu(r)\,dx\leq$
$\displaystyle\qquad\leq\int_{s}^{r}\\!\\!\\!\int_{M}\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla(u_{k})_{h}-\frac{1}{2}|\nabla(u_{k})_{h}|^{2}\,\mu\,dxd\tau$
$\displaystyle\qquad\quad+\varepsilon\int_{s}^{r}\\!\\!\\!\int_{M}\frac{1}{h}\int_{\tau}^{\tau+h}\mathds{1}_{\\{u\geq-k\\}}\left(\log
m+V\right)\mu(\tau)dsdxd\tau\,.$

Now we let $h\to 0$. Once more, last term can be handled through Fatou’s
lemma; indeed, since $\log(m)\in L^{1}_{loc}((0,T);L^{1}(M))$, we have that

$\frac{1}{h}\int_{\tau}^{\tau+h}\mathds{1}_{\\{u\geq-k\\}}\left(\log
m+V\right)\mathop{\to}\limits^{h\to 0}\mathds{1}_{\\{u\geq-k\\}}\left(\log
m+V\right)\quad\hbox{for a.e. $\tau\in(0,T),x\in M$}$

and

$\displaystyle\mu(\tau)\frac{1}{h}\int_{\tau}^{\tau+h}\mathds{1}_{\\{u\geq-k\\}}\left(\log
m+V\right)\leq\mu\,\log\mu+\mu\,V+m_{h}$

where last sequence is strongly convergent in $L^{1}(Q_{T})$. Hence Fatou’s
lemma can be applied and yields

$\displaystyle\limsup_{h\to
0}\int_{s}^{r}\\!\\!\\!\int_{M}\frac{1}{h}\int_{\tau}^{\tau+h}\mathds{1}_{\\{u\geq-k\\}}\left(\log
m+V\right)\mu(\tau)dsdxd\tau$
$\displaystyle\qquad\leq\int_{s}^{r}\\!\\!\\!\int_{M}\mathds{1}_{\\{u\geq-k\\}}\left(\log
m+V\right)\mu dxd\tau\,.$

Similarly we argue for the term involving $\nabla u_{k}$, where we also use
Fatou’s lemma, because $\nabla u_{k}\in L^{2}_{loc}((0,T)\times M)$ and
$\nabla(u_{k})_{h}\to\nabla u_{k}$ almost everywhere, up to extracting a
suitable subsequence. Finally, using that $u_{k}$ is uniformly bounded in
$(0,r+h)$ and $u$ admits one-sided traces (in the sense of monotone limits of
measurable functions, as recalled above), we have that $(u_{k})_{h}(t)\to
u_{k}(t)$ (we can use here the precise representative for $u$ at any $t$,
otherwise we should limit ourselves to a.e. $t$). Notice that the convergence
$(u_{k})_{h}(t)\to u_{k}(t)$ is pointwise but also weak$-*$ $L^{\infty}$, and
holds for all $t\geq 0$. Therefore, once $h\to 0$ we get

$\displaystyle\int_{M}u_{k}(s)\mu(s)\,dx-\int_{M}u_{k}(r)\,\mu(r)\,dx$
$\displaystyle\leq\int_{s}^{r}\\!\\!\\!\int_{M}[\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla
u_{k}-\frac{1}{2}|\nabla u_{k}|^{2}\,\mu]\,dxd\tau$
$\displaystyle+\varepsilon\int_{s}^{r}\\!\\!\\!\int_{M}\mathds{1}_{\\{u\geq-k\\}}\left(\log
m+V\right)\mu dxd\tau\,.$

Letting now $k\to\infty$, using Fatou’s lemma (and the monotone convergence
theorem if $s=0$), we obtain

$\displaystyle\int_{M}u(s)\mu(s)\,dx-\int_{M}u(r)\,\mu(r)\,dx$
$\displaystyle\leq\int_{s}^{r}\\!\\!\\!\int_{M}[\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla
u-\frac{1}{2}|\nabla u|^{2}\,\mu]\,dxd\tau$
$\displaystyle\quad+\varepsilon\int_{s}^{r}\\!\\!\\!\int_{M}\left(\log
m+V\right)\mu dxd\tau\,.$

With a symmetric argument, using the left time-averages $u_{-h}$, we also
obtain the inequality

$\displaystyle\int_{M}u(r)\mu(r)\,dx-\int_{M}u(t)\,\mu(t)\,dx$
$\displaystyle\leq\int_{r}^{t}\\!\\!\\!\int_{M}[\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla
u-\frac{1}{2}|\nabla u|^{2}\,\mu]\,dxd\tau$
$\displaystyle\quad+\varepsilon\int_{r}^{t}\\!\\!\\!\int_{M}\left(\log
m+V\right)\mu dxd\tau\,$

for every $0<r<t\leq T$. Adding the last two inequalities we obtain (5.14). ∎

######  Remark 5.8:

The inequality (5.14) includes the case that $s=0$ or $t=T$. In particular, it
is a byproduct of the previous proof that $u(0)\in L^{1}(dm_{0})$ and $u(T)\in
L^{1}(dm_{1})$, which would not be guaranteed a priori. This is indeed a
consequence of item 2 of the statement, which implies that
$\int_{M}u(0)m_{0}\,dx$ is bounded below and $\int_{M}u(T)m_{1}\,dx$ is
bounded above. Since the other bounds are obvious from (5.4), this yields
$u(0)\in L^{1}(dm_{0}),u(T)\in L^{1}(dm_{1})$.

######  Remark 5.9:

We point out that inequality (5.14) remains true when $\varepsilon=0$ even
without requiring that $\mu\log(\mu)\in L^{1}(Q_{T})$. It is enough that
$\mu\in L^{1}(Q_{T})$ in order that (5.14) holds for all $s,t\in(0,T)$ (such
that $\mu(s),\mu(t)\in L^{1}(M)$), and even for $s=0,t=T$ assuming for
instance that $\mu(\cdot)$ is continuous in $[0,T]$ in the weak
$L^{1}$-topology.

In fact, we know from Proposition 4.4 that $\varepsilon_{n}(\log(m_{n}))_{+}$
is locally uniformly bounded; in addition, if $\varepsilon_{n}\to 0$, using
estimate (3.9) we have

$\displaystyle\|\varepsilon_{n}(\log(m_{n}))_{+}\|_{L^{1}((0,T)\times M)}$
$\displaystyle\leq\varepsilon_{n}\int_{0}^{T}\\!\\!\\!\\!\int_{M}m_{n}(\log(m_{n}))_{+}$
$\displaystyle\leq C\,\varepsilon_{n}\left(1+|\log(\varepsilon_{n})|\right)\to
0\,.$

Hence $\varepsilon_{n}(\log(m_{n}))_{+}$ converges to zero in $L^{1}$ and
weakly$-*$ in $L^{\infty}((a,b)\times M)$. This implies that last term in
(5.18) vanishes as $n\to\infty$, only using that $\mu$ is in $L^{1}(Q_{T})$.
Thus we obtain again the inequality

$\int_{M}u_{k}(s)\mu(s)\,dx-\int_{M}u_{k}(r)\,\mu(r)\,dx\leq\int_{s}^{r}\\!\\!\\!\int_{M}[\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla
u_{k}-\frac{1}{2}|\nabla u_{k}|^{2}\,\mu]\,dxd\tau$

for any $0<s<r<T$. Since $u$ is locally bounded, here one can readily get rid
of the truncation $k$ for $s,r\in(0,T)$. To get the inequality up to $t=0$,
one can first let $s\to 0^{+}$ using that $\mu(\cdot)$ is continuous in the
weak $L^{1}$-topology and $u_{k}$ is uniformly bounded. This leads the first
integral towards $\int_{M}u_{k}(0)m_{0}\,dx$, which gives the desired term by
letting $k\to\infty$ and using the monotone convergence theorem. Simmetrically
one argue up to $t=T$ to get (5.14) in the whole interval $(0,T)$. ∎

Now we have all the ingredients to prove our main result on the existence and
characterization of minima for nonnegative marginals $m_{0},m_{1}$, which are
only assumed to be $L^{1}(M)$.

######  Theorem 5.10:

Let $V\in W^{2,\infty}(M),\,\,m_{0},m_{1}\in\mathcal{P}(M)\cap L^{1}(M)$, and
$\varepsilon>0$.

Then there is a unique $m$ and a unique $u$ such that $(u,m)$ is a weak
solution of problem (5.2) with $\int_{M}u(T)m_{1}=0$. Moreover we have that
$u,m\in L^{\infty}_{loc}(Q_{T})$, and $(m,\nabla u)$ is the unique minimum of
the functional $\mathcal{F}_{\varepsilon}$ in (1.1).

###### Proof.

We first approximate $m_{0},m_{1}$ with positive smooth marginals. To this
goal, we consider the solutions of the heat equation

$\frac{\partial}{\partial t}\tilde{m}_{i}=\Delta\tilde{m}_{i}\quad\text{with
$\tilde{m}_{i}(0,\cdot)=m_{i}$ for $i=0,1$.}$

Thanks to the compactness of $M$, it is well-known that such solutions are
smooth on $(0,+\infty)\times M$ and that they are curves of probability
measures (cfr. [21, Chapter 7]). Furthermore we have
$\tilde{m}_{i}(t,\cdot)>0$ for every $t>0$ by the strong parabolic maximum
principle (cfr. [21, Theorem 8.11]) and $\tilde{m}_{i}(t,\cdot)\to m_{i}$ in
$L^{1}(M)$ for $t\to 0$ ([21, Theorem 7.19]).

For every $n\in\mathbb{N}$ let $m_{0,n}=\tilde{m}_{0}(\frac{1}{n},\cdot)$ and
$m_{1,n}=\tilde{m}_{1}(\frac{1}{n},\cdot)$. By Theorem 5.4, we obtain the
existence of a couple $(u_{n},m_{n})$ which is a classical solution of

$\begin{cases}-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}=\varepsilon(\log(m)+V(x))\quad&\text{in $Q_{T}$}\\\
\partial_{t}m-div_{g}(m\nabla u)=0&\text{in $Q_{T}$}\\\
m(0,\cdot)=m_{0,n},\quad m(T,\cdot)=m_{1,n}&\text{in $M$}\end{cases}$ (5.19)

with $\int_{M}u_{n}(T)m_{1,n}=0$ $\forall n\in\mathbb{N}$.

Now we use Lemma 5.2 and Lemma 5.5 to get estimates for $u_{n},m_{n}$. In
particular, $u_{n}$ satisfies (5.4), hence it is locally uniformly bounded,
and the same holds for $m_{n}$ due to Corollary 5.3. In addition $u_{n}$ is
bounded in $L^{2}((a,b);H^{1}(M))$ and $m_{n}|\nabla u_{n}|^{2}$ is bounded in
$L^{1}(Q)$. With the same compactness arguments used in Lemma 5.6, we deduce
that both $u_{n}$ and $m_{n}$ are relatively compact in the strong
$L^{1}(M)$-topology, locally in time. Therefore, we deduce that there exist
functions $u,m$ such that for a subsequence, not relabeled,

$\displaystyle u_{n}$ $\displaystyle\rightarrow u\quad\hbox{weakly in
$L^{2}((a,b);H^{1}(M))$ and strongly in $L^{p}((a,b)\times(M))\,\forall
p>1$,}$ $\displaystyle m_{n}$ $\displaystyle\rightarrow m\quad\hbox{strongly
in $L^{p}((a,b)\times(M))\,\forall p>1$, and a.e. in $Q$.}$

In addition, we know that $m_{n}(t)$ has bounded entropy (at any time $t$), so
it is weakly compact in $L^{1}(M)$; and due to the bound of $m_{n}|\nabla
u_{n}|^{2}$ (see (5.5)), we get that $m_{n}$ is equi-continuous from $[0,T]$
into the space of measures. By Ascoli-Arzela’s theorem, $m_{n}(t)\to m(t)$ in
the Wasserstein topology (uniformly in $[0,T]$), and actually in $L^{1}$-weak
for all $t\in(0,T)$, due to the bound of the entropy. By continuity, we
conclude that $m(0)=m_{0}$ and $m(T)=m_{1}$. Notice that, by the local strong
convergence of $m_{n}$, and due to the bound (5.5), we also deduce that

$\sqrt{m_{n}}\nabla u_{n}\to\sqrt{m}\nabla u\quad\hbox{weakly in
$L^{2}((0,T)\times M))$,}$ (5.20)

and, in particular, we have that $m_{n}\nabla u_{n}\to m\nabla u$ in the sense
of distributions, and actually weakly in $L^{1}((0,T)\times M)$. Thus, we
proved so far that $m\in C^{0}([0,T];{\mathcal{P}}(M))$ and is a weak solution
of

$\begin{cases}\partial_{t}m-div_{g}(m\nabla u)=0&\hbox{in $Q_{T}$,}\\\
m(0)=m_{0}\,,\,m(T)=m_{1}\,.&\end{cases}$ (5.21)

In addition, by (5.6) and Fatou’s lemma, we have that $\log(m)\in
L^{1}((a,b)\times(M))$, for any $0<a<b<T$, and in particular $m>0$ a.e. in
$Q$.

As for the Hamilton-Jacobi equation, we use Lemma 5.6 to deduce that

$-\partial_{t}u+\frac{1}{2}|\nabla u|^{2}\leq\varepsilon(\log(m)+V)\,.$ (5.22)

and we also have

$\limsup_{n\to\infty}\int_{M}m_{0n}u_{n}(0)dx\leq\int u(0)dm_{0}\,.$ (5.23)

In particular (since the upper bound follows from (5.4)), we deduce that
$u(0)\in L^{1}(dm_{0})$. Similarly we reason for $t=T$, obtaining

$\int_{M}m_{1}\,u(T)\,dx\leq\liminf_{n\to\infty}\int_{M}m_{1n}u_{n}(T)=0\,.$
(5.24)

As before, this implies, in particular, that $u(T)\in L^{1}(dm_{1})$. Now we
only need to conclude that $(u,m)$ satisfy condition (iii) of Definition 5.1.
To this purpose, we follow the steps of [41], [12], on account of Lemma 5.6.
First we go back to (5.19), which implies, using $\int_{M}u_{n}(T)m_{1n}=0$,

$\int_{M}m_{0n}\hat{u}_{n}(0)dx=\int_{0}^{T}\\!\\!\int_{M}\frac{1}{2}m_{n}|\nabla
u_{n}|^{2}+\varepsilon m_{n}(\log m_{n}+V)\,dxdt\,.$

Using (5.20) and weak lower semicontinuity, as well as (5.23), we deduce, as
$n\to\infty$:

$\int_{0}^{T}\\!\\!\int_{M}\frac{1}{2}m|\nabla u|^{2}+\varepsilon m(\log
m+V)\,dxdt\leq\int u(0)dm_{0}\,.$ (5.25)

However, applying (5.14) with $(\mu,v)=(m,\nabla u)$, $s=0$ and $t=T$, we get

$\displaystyle\int_{M}m_{0}u(0)dx$
$\displaystyle\leq\int_{0}^{T}\\!\\!\int_{M}\frac{1}{2}\left|\nabla
u\right|^{2}m+\varepsilon m(\log m+V)\,dxdt+\int_{M}u(T)\,dm_{1}$
$\displaystyle\leq\int_{0}^{T}\\!\\!\int_{M}\frac{1}{2}\left|\nabla
u\right|^{2}m+\varepsilon m(\log m+V)\,dxdt$

where we used (5.24). Putting together the above information with (5.25) we
obtain

$\int_{0}^{T}\\!\\!\int_{M}\frac{1}{2}m|\nabla u|^{2}+\varepsilon m(\log
m+V)\,dxdt=\int_{M}m_{0}u(0)dx\,,$

and, in between, we also get that $\int_{M}m_{1}\,u(T)\,dx=0$. This means that
$(u,m)$ satisfy Definition 5.1. In addition, from the bounds derived before,
we have $u,m\in L^{\infty}_{loc}(Q_{T})$.

We are left to prove that $(m,\nabla u)$ is the unique minimum of
$\mathcal{F}_{\varepsilon}$. To show that, let $(\mu,v)$ be an admissible
couple for the functional ${\mathcal{F}}_{\varepsilon}$. Without loss of
generality, we can assume that $\mathcal{F}_{\varepsilon}(\mu,v)<\infty$, in
particular $\mu\log\mu\in L^{1}(Q_{T})$. We use once more (5.25) together with
(5.14) and we get

$\displaystyle\int_{0}^{T}\\!\\!\int_{M}$
$\displaystyle\left(\frac{1}{2}m|\nabla u|^{2}+\varepsilon m(\log
m+V)\right)dxdt\leq\int_{0}^{T}\\!\\!\\!\int_{M}[\mu\,v\,\operatorname{\cdot_{g}\\!}\nabla
u-\frac{1}{2}|\nabla u|^{2}\,\mu]\,dxdt$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad+\varepsilon\int_{0}^{T}\\!\\!\\!\int_{M}\left(\log
m+V\right)\mu dxdt$
$\displaystyle\qquad\qquad\qquad\qquad\leq\int_{0}^{T}\\!\\!\\!\int_{M}\frac{1}{2}|v|^{2}\,\mu\,dxdt+\varepsilon\int_{0}^{T}\\!\\!\\!\int_{M}\left(\log
m+V\right)\mu dxdt\,.$

By the strict convexity of $r\to r\log r-r$, for every $a\geq 0$ and $b>0$ we
have

$a\log(a)-a\geq b\log(b)-b+\log(b)(a-b)$

where $a\log a$ is extended as $0$ for $a=0$. Furthermore the equality holds
if and only if $a=b$. This is equivalent to

$(\log(b)-\log(a))a\leq b-a$ (5.26)

with equality if and only if $a=b$. Applying this inequality with $a=\mu$ and
$b=m$ (which is positive a.e.), we obtain

$\displaystyle\int_{0}^{T}\\!\\!\int_{M}\left(\frac{1}{2}m|\nabla
u|^{2}+\varepsilon m(\log m+V)\right)dxdt$
$\displaystyle\leq{\mathcal{F}}_{\varepsilon}(\mu,v)+\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}(m-\mu)dxdt$
$\displaystyle=\,{\mathcal{F}}_{\varepsilon}(\mu,v)$

and the equality holds if and only if $\mu=m$. This concludes the proof that
$(m,\nabla u)$ is the unique minimum of $\mathcal{F}_{\varepsilon}$.

In fact, the solution we found is also the unique weak solution of the system
(5.2) (up to addition of a constant to $u$). Indeed, the uniqueness of $(m,u)$
can be proved similarly as in [12, Thm 1.16]. Compared to this latter result,
we observe that, being $m>0$ almost everywhere, there is no loss of
information here due to the set where $m$ vanishes. ∎

##  6 Convergence to Wasserstein geodesic

We now briefly analyze the limit $\varepsilon\rightarrow 0$ to show that the
minimal curves of $\mathcal{F}_{\varepsilon}$ converge to the geodesics of the
classical mass transport problem

$\min\mathcal{F}_{0}(m,v)\coloneqq\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}\left|v\right|^{2}\,dm\,,\quad(m,v):\begin{cases}\partial_{t}m-div_{g}(vm)=0\\\
m(0)=m_{0}\\\ m(T)=m_{1}\,.\end{cases}$ (6.1)

We first consider the easier case that the marginals $m_{0},m_{1}$ have finite
entropy. This assumption implies that $\mathcal{F}_{\varepsilon}$ converges to
$\mathcal{F}_{0}$ with a rate of order $\varepsilon$.

######  Theorem 6.1:

Let $V\in W^{2,\infty}(M),\,\,m_{0},m_{1}\in\mathcal{P}(M)\cap L^{1}(M)$ and
such that ${\mathcal{H}}(m_{0};\nu),{\mathcal{H}}(m_{1};\nu)<\infty$. For
$\varepsilon\in(0,1)$, let $(m_{\varepsilon},u_{\varepsilon})$ be the unique
solution of (5.2) given by Theorem 5.10, and $(m_{\varepsilon},\nabla
u_{\varepsilon})$ the unique minima of $\mathcal{F}_{\varepsilon}$.

As $\varepsilon\to 0$, we have:

$\begin{split}m_{\varepsilon}&\to m\quad\hbox{in
$C^{0}([0,T],{\mathcal{P}}(M))$ and weakly in $L^{1}(Q_{T})$,}\\\
m_{\varepsilon}\nabla u_{\varepsilon}&\to m\nabla u\quad\hbox{weakly in
$L^{1}(Q_{T})$,}\end{split}$ (6.2)

where $m$ is the Wasserstein geodesic between $m_{0},m_{1}$, and $(m,\nabla
u)$ is a minimum of $\mathcal{F}_{0}$.

Moreover, we have $(\min\mathcal{F}_{0})=\lim\limits_{\varepsilon\rightarrow
0}(\min\mathcal{F}_{\varepsilon})$, and in particular, for some $K>0$,

$|\min\mathcal{F}_{\varepsilon}-\min\mathcal{F}_{0}|\leq K\,\varepsilon\,.$
(6.3)

###### Proof.

For every $\varepsilon>0$ we can apply Theorem 5.10 and define the couple
$(u_{\varepsilon},m_{\varepsilon})$ which is the unique weak solution to the
problem

$\begin{cases}-\partial_{t}u+\frac{1}{2}|\nabla
u|^{2}=\varepsilon(\log(m)+V(x))\quad&\text{in $Q_{T}$}\\\
\partial_{t}m-div_{g}(m\nabla u)=0&\text{in $Q_{T}$}\\\ m(0)=m_{0},\quad
m(T)=m_{1}&\text{in $M$}\end{cases}$

with $\int_{M}u_{\varepsilon}(T)m_{1}=0$.

By Lemma 5.2, we have that $u_{\varepsilon}$ is bounded in
$L^{2}((a,b),H^{1}(M))$ and in $L^{\infty}((a,b)\times M)$ for every
$0<a<b<T$, so it is weakly relatively compact in $L^{2}((a,b);H^{1}(M))$.

For any sequence extracted out of $u_{\varepsilon}$, by a diagonal process we
can select (and fix) a function $u\in L^{2}_{loc}((0,T);H^{1}(M))\cap
L^{\infty}_{loc}((0,T)\times M)$ and a subsequence (that we will not rename)
such that $u_{\varepsilon}$ converges weakly to $u$ in $L^{2}((a,b);H^{1}(M))$
for every $0<a<b<T$.

Furthermore, as a consequence of estimate (5.5), we have
$d_{W}(m_{\varepsilon}(t),m_{\varepsilon}(s))\leq C\sqrt{t-s}$ for any $t>s$
and some $C>0$, where $d_{W}$ is the Wasserstein distance. Hence, by Ascoli-
Arzela’s theorem, there exists $m\in C([0,T];{\mathcal{P}}(M))$ such that, up
to a subsequence, $m_{\varepsilon}(t)\to m(t)$ in the Wasserstein topology,
uniformly in time. Since $m_{0},m_{1}$ have finite entropy, by estimate (3.8),
we have that $\int_{M}m_{\varepsilon}(t)\log(m_{\varepsilon}(t))$ is uniformly
bounded in $(0,T)$. We deduce that $m_{\varepsilon}$ is weakly relatively
compact in $L^{1}(Q_{T})$ and then, by Schwartz inequality and (5.5),
$m_{\varepsilon}\nabla u_{\varepsilon}$ is also weakly relatively compact in
$L^{1}(Q_{T})$. In particular, there exists $m\in C([0,T];{\mathcal{P}}(M))$,
and $w\in L^{1}(Q_{T})$, such that

$\displaystyle m_{\varepsilon}$ $\displaystyle\to m\qquad\hbox{in
$C([0,T];{\mathcal{P}}(M))$ and weakly in $L^{1}(Q_{T})$,}$ $\displaystyle
m_{\varepsilon}\nabla u_{\varepsilon}$ $\displaystyle\to w\qquad\hbox{ weakly
in $L^{1}(Q_{T})$.}$

We now identify $w$ as $m\nabla u$. To this goal, we first use the semi-
continuity for the function $\Psi$ defined in (3.2), and we get

$\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{|w|^{2}}{2m}dxdt\leq\liminf_{\varepsilon\to
0}\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m_{\varepsilon}|\nabla
u_{\varepsilon}|^{2}dxdt=\int_{M}u_{\varepsilon}(0)\,m_{0}\,dx+O(\varepsilon)$
(6.4)

where we used the bound on the entropy. By Lemma 5.6 we deduce

$\begin{split}\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{|w|^{2}}{2m}dxdt&\leq\liminf_{\varepsilon\to
0}\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m_{\varepsilon}|\nabla
u_{\varepsilon}|^{2}dxdt\\\ &\leq\limsup_{\varepsilon\to
0}\int_{M}u_{\varepsilon}(0)\,m_{0}\,dx\leq\int_{M}u(0)m_{0}\,dx\,.\end{split}$
(6.5)

Notice that this inequality also implies that $w=0$ a.e. in the set
$\\{m=0\\}$. Setting $v:=\frac{w}{m}\mathds{1}_{\\{m>0\\}}$ we deduce that
$(m,v)$ is a solution to (2.3). We also get from Lemma 5.6 a similar
inequality at $t=T$, namely

$\int_{M}u(T)m_{1}\,dx\leq\liminf_{\varepsilon\to
0}\int_{M}u_{\varepsilon}(T)\,m_{1}\,dx=0\,.$

We insert this information into (5.14) (where $\varepsilon=0,s=0,t=T$) and we
get

$\int_{M}u(0)m_{0}\,dx\leq\int_{0}^{T}\\!\\!\\!\\!\int_{M}[\frac{w}{m}\operatorname{\cdot_{g}\\!}\nabla
u-\frac{|\nabla
u|^{2}}{2}]\,m\,dx\leq\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{|w|^{2}}{2m}\,dxdt\,.$

Combining this with (6.5) we conclude that $w=m\,\nabla u$ and that

$\int_{M}u(0)m_{0}\,dx=\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m|\nabla
u|^{2}dxdt\,,$

and then

$\int_{M}u_{\varepsilon}(0)\,m_{0}\,dx\to\int_{M}u(0)m_{0}\,dx\,,$

and

$\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m_{\varepsilon}|\nabla
u_{\varepsilon}|^{2}dxdt\to\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m|\nabla
u|^{2}dxdt\,.$

We now show that $(m,\nabla u)$ is a minimum of $\mathcal{F}_{0}$. To this
goal, we recall (see [14], [38]) that the minimum of $\mathcal{F}_{0}$ is
attained at a unique geodesic $\mu^{*}$ such that $\mu^{*}(t)\in L^{1}(M)$ for
every $t$ and $\mu^{*}(\cdot)$ is continuous in the weak-$L^{1}$ topology. On
account of Remark 5.9, we can use inequality (5.14) with $\varepsilon=0$ and
$\mu=\mu^{*}$, which yields:

$\displaystyle\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m|\nabla u|^{2}dxdt$
$\displaystyle=\int_{M}u(0)m_{0}\,dx\leq\int_{0}^{T}\\!\\!\\!\int_{M}[\mu^{*}\,v\,\operatorname{\cdot_{g}\\!}\nabla
u-\frac{1}{2}|\nabla u|^{2}\,\mu^{*}]\,dxdt$
$\displaystyle\leq\int_{0}^{T}\\!\\!\\!\int_{M}\frac{1}{2}|v|^{2}\,\mu^{*}\,dxdt=\min\mathcal{F}_{0}$

Hence $(m,\nabla u)$ is the minimum point of $\mathcal{F}_{0}$ and coincides
with the unique geodesic between $m_{0},m_{1}$ (notice that the previous
inequality implies $v=\nabla u$). Finally, we observe that, still using (5.14)
for $(m_{\varepsilon},u_{\varepsilon})$ and $(m,u)$, we have

$\displaystyle\min\mathcal{F}_{\varepsilon}$
$\displaystyle=\int_{M}u_{\varepsilon}(0)\,m_{0}\,dx$
$\displaystyle\leq\int_{0}^{T}\\!\\!\\!\\!\int_{M}\frac{1}{2}m|\nabla
u|^{2}dxdt+\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}m(\log(m_{\varepsilon})+V)\,dxdt$
$\displaystyle\leq\min\mathcal{F}_{0}+\varepsilon\int_{0}^{T}\\!\\!\\!\\!\int_{M}m\log(m)+C\,\varepsilon=\min\mathcal{F}_{0}+O(\varepsilon)$

due to the fact that $m$ has finite entropy. Since the opposite inequality is
obviously true, we conclude with (6.3). ∎

As a byproduct of the previous result, we have proved that whenever
$m_{0},m_{1}$ have finite entropy, then the Wasserstein geodesic connecting
$m_{0},m_{1}$ has finite entropy for all times. In fact, using Corollary 3.4
we also have a quantitative estimate of the semiconvexity of the log-entropy
functional along the geodesic. In this way, we recover a result proved by
Daneri and Savaré ([15]) with a different and independent approach. We point
out that we can avoid the use of this semiconvexity property of the geodesic
in all our estimates and stability results (see also Remark 2.4), so that this
is just deduced from the limit $\varepsilon\to 0$ of the semiconvexity of the
optimal curves of $\mathcal{F}_{\varepsilon}$.

######  Corollary 6.2:

In the assumptions of Theorem 6.1, let $\Lambda\in\mathbb{R}$ be such that

$Ric_{g}+D^{2}V\geq\Lambda\,I_{g}$

in the sense that $(Ricc_{g}+D^{2}V)(X,X)\geq\Lambda\left|X\right|^{2}$ for
every vector field $X$ on $M$.

Then the relative entropy functional ${\mathcal{H}}(m;\nu)$ is
$\Lambda$-convex along the 2-Wasserstein geodesics. In other words, if
$m:[0,1]\to{\mathcal{P}}(M)$ is the geodesic between $m_{0}$ and $m_{1}$, it
holds

${\mathcal{H}}(m(t);\nu)\leq
t{\mathcal{H}}(m_{1};\nu)+(1-t){\mathcal{H}}(m_{0};\nu)-\frac{\Lambda}{2}t(1-t)W_{2}^{2}(m_{0},m_{1})$
(6.6)

for every $t\in[0,1]$.

###### Proof.

As in Theorem 6.1, let $(u_{\varepsilon},m_{\varepsilon})$ be the sequence of
solutions of (1.2), with $\int_{M}u_{\varepsilon}(T)m_{1}=0$. At first, let us
suppose that $m_{0},m_{1}$ are smooth and positive, so that
$u_{\varepsilon},m_{\varepsilon}$ are smooth solutions (Theorem 5.4). By (3.7)
in Corollary 3.4, we have

$\displaystyle\frac{d^{2}}{dt^{2}}{\mathcal{H}}(m_{\varepsilon}(t);\nu)$
$\displaystyle\geq\Lambda\int_{M}m_{\varepsilon}\,|\nabla
u_{\varepsilon}|^{2}dx$
$\displaystyle=2\Lambda{\mathcal{B}}_{\varepsilon}(m_{0},m_{1})+2\Lambda\varepsilon{\mathcal{H}}(m_{\varepsilon}(t);\nu)$

where we used (3.4) (with $T=1$) and, we recall,
${\mathcal{B}}_{\varepsilon}(m_{0},m_{1})=\min(\mathcal{F}_{\varepsilon})$.
Since $m_{0},m_{1}$ have finite entropy, we already know from (3.8) that
${\mathcal{H}}(m_{\varepsilon}(t);\nu)$ is bounded independently of
$\varepsilon$, for every $t\in(0,T)$. Hence we get, for some $C>0$:

$\frac{d^{2}}{dt^{2}}{\mathcal{H}}(m_{\varepsilon}(t);\nu)\geq\Lambda_{\varepsilon}:=2\Lambda{\mathcal{B}}_{\varepsilon}(m_{0},m_{1})-C\,\varepsilon\,$

which implies

${\mathcal{H}}(m_{\varepsilon}(t);\nu)\leq
t{\mathcal{H}}(m_{1};\nu)+(1-t){\mathcal{H}}(m_{0};\nu)-\frac{\Lambda_{\varepsilon}}{2}t(1-t)\quad\forall
t\in([0,1].$ (6.7)

Note that $\Lambda_{\varepsilon}$ is stable by approximation of $m_{0},m_{1}$
with smooth positive densities, due to Theorem 5.10, so the above inequality
holds for any $m_{0},m_{1}$ with finite entropy. Finally, we let
$\varepsilon\to 0$; by Theorem 6.1 we know that $m_{\varepsilon}(t)$ weakly
converges in $L^{1}(M)$ towards $m(t)$, where $m$ is the Wasserstein geodesic
between $m_{0},m_{1}$. Thus we can use the weak lower semicontinuity of the
entropy for the left-hand side. We also know that
${\mathcal{B}}_{\varepsilon}(m_{0},m_{1})=\min(\mathcal{F}_{\varepsilon})$
converges towards $\min(\mathcal{F}_{0})=\frac{1}{2}W_{2}^{2}(m_{0},m_{1})$.
Hence $\Lambda_{\varepsilon}\to\Lambda\,W_{2}^{2}(m_{0},m_{1})$ and from (6.7)
we deduce (6.6). ∎

We conclude by extending the convergence result of Theorem 6.1 to marginals
which only belong to $L^{1}(M)$, without having necessarily finite entropy. It
is known that, if $m_{0},m_{1}\in L^{1}(M)$, then the Wasserstein geodesic
belongs to $L^{1}(M)$ for all times $t$, see e.g. [14], [38]. This is also a
byproduct of our next result, since we will prove that (6.2) still holds for
merely $L^{1}$ marginals. To get the necessary equi-integrability for this
goal, we will use an idea suggested to us by G. Savaré, based on displacement
convexity and the following lemma essentially contained in [49].

######  Lemma 6.3:

Let $m_{0}\in L^{1}(M)$. Then there exists a function
$U:[0,\infty)\to\mathbb{R}^{+}$ such that:

(i) $U\in C^{2}(0,\infty)$, is convex and satisfies
$\frac{U(r)}{r}\mathop{\to}\limits^{r\to\infty}+\infty$.

(ii) $P^{\prime}(r)r-(1-\frac{1}{d})P(r)\geq 0$, and $P(r)\leq K\,r$ for every
$r>0$, where $P(r)=U^{\prime}(r)r-U(r)$, $K>0$.

(iii) $U(m_{0})\in L^{1}(M)$.

Even if the above Lemma mostly follows from [49, Proposition 17.7] combined
with De la Vallée Poussin lemma, we will give a proof in the Appendix, for the
reader’s convenience. Standing on Lemma 6.3, we will show that
$\min\mathcal{F}_{\varepsilon}(m_{0},m_{1})\to\min\mathcal{F}_{0}(m_{0},m_{1})$,
although now the rate of convergence appears to be of order
$O(\varepsilon\log\varepsilon)$. We will also prove a further property here,
namely that, up to approximating $m_{0},m_{1}$ with suitable smooth sequences
$m_{0\varepsilon},m_{1\varepsilon}$, we can build a minimizing curve of
$\mathcal{F}_{\varepsilon}$ which is a smooth approximation of the Wasserstein
geodesic, with the adjoint states uniformly converging to the Kantorovich
potentials.

######  Theorem 6.4:

Let $V\in W^{2,\infty}(M)$, and $m_{0},m_{1}\in\mathcal{P}(M)\cap L^{1}(M)$.
For $\varepsilon\in(0,1)$, let $(m_{\varepsilon},u_{\varepsilon})$ be the
unique solution of (5.2) given by Theorem 5.10, and $m$ be the Wasserstein
geodesic between $m_{0},m_{1}$, with $(m,\nabla u)$ the minimum of
$\mathcal{F}_{0}$. Then we have that (6.2) holds true and
$\min\mathcal{F}_{0}=\lim\limits_{\varepsilon\rightarrow
0}\min\mathcal{F}_{\varepsilon}$. In particular, we have

$\min\mathcal{F}_{0}-c_{0}\varepsilon\leq\min\mathcal{F}_{\varepsilon}\leq\min\mathcal{F}_{0}+c_{1}\,\varepsilon|\log\varepsilon|\,$
(6.8)

for some $c_{0},c_{1}>0$.

In addition, there exist sequences $m_{0\varepsilon},m_{1\varepsilon}$,
converging respectively to $m_{0},m_{1}$ in $L^{1}(M)$, such that:

(i) $(m_{\varepsilon},\nabla u_{\varepsilon}):={\rm
argmin}\mathcal{F}_{\varepsilon}$ is smooth in $(0,T)\times M$.

(ii) $u_{\varepsilon}$ is bounded in $W^{1,\infty}(Q_{T})$ and converges
uniformly to a Lipschitz continuous solution $u$ of the Hamilton-Jacobi
equation $\partial_{t}u=|\nabla u|^{2}/2$ in $Q_{T}$.

(iii) $m_{\varepsilon}\to m$ in $C^{0}([0,T],{\mathcal{P}}(M))$ where $m$ is
the Wasserstein geodesic connecting $m_{0},m_{1}$, with $\nabla u$ being the
metric velocity of the geodesic and $u(0),u(T)$ the Kantorovich optimal
potentials.

###### Proof.

Let $U$ be the function given by Lemma 6.3 (built replacing $m_{0}$ with
$\max(m_{0},m_{1})$). Using Proposition 3.3, we have

$\displaystyle\frac{d^{2}}{dt^{2}}\int_{M}U(m_{\varepsilon})\,dx$
$\displaystyle\geq\int_{M}P(m_{\varepsilon})Ricc_{g}(\nabla
u_{\varepsilon},\nabla
u_{\varepsilon})\,dx-\varepsilon\int_{M}m_{\varepsilon}\,|\nabla V|^{2}\,dx$
$\displaystyle\geq-\lambda\,K\int_{M}m_{\varepsilon}\,|\nabla
u_{\varepsilon}|^{2}\,dx-c\,\varepsilon\,$

where we used the property (ii) of $U$, from Lemma 6.3. Setting
$\varphi_{\varepsilon}(t)=\int_{M}U(m_{\varepsilon})(t)\,dx$, we deduce that
$\varphi_{\varepsilon}$ satisfies

$\begin{cases}-\varphi_{\varepsilon}^{\prime\prime}(t)\leq\lambda\,Kf_{\varepsilon}(t)+c\varepsilon&t\in(0,T)\\\
\varphi_{\varepsilon}(0)=\int_{M}U(m_{0})\,,\varphi_{\varepsilon}(T)=\int_{M}U(m_{1})&\,,\end{cases}$

where $f_{\varepsilon}:=\int_{M}m_{\varepsilon}\,|\nabla
u_{\varepsilon}|^{2}\,dx$ is bounded in $L^{1}(0,T)$ by Lemma 5.2. By the
(compact) embedding of $H^{1}(0,T)$ into $C^{0}([0,T])$, we deduce that
$\varphi_{\varepsilon}$ is uniformly bounded and actually it is relatively
compact in the uniform topology. Since $U$ is superlinear, this means that
$m_{\varepsilon}(t)$ is weakly compact in $L^{1}(M)$ and weakly converges to
$m(t)$, for every $t\in[0,T]$. In addition, $t\mapsto m(t)$ is continuous in
$L^{1}(M)$ endowed with the weak topology. With this in hand, we also have
that $m_{\varepsilon}\nabla u_{\varepsilon}$ is weakly compact in
$L^{1}(Q_{T})$. Moreover, from Lemma 5.2, we know that $u_{\varepsilon}$
weakly converges to some $u\in L^{2}_{loc}((0,T);H^{1}(M))\cap
L^{\infty}_{loc}((0,T)\times M)$, exactly as in Theorem 6.1. In order to
identify the limit of $m_{\varepsilon}\nabla u_{\varepsilon}$ as $m\nabla u$,
we can proceed as before, using (5.14) on account of Remark 5.9. Thus we
obtain the same conclusion (6.2) as before. However, the rate of convergence
(6.3) does not follow in this case since we can only estimate

$\int_{0}^{T}\\!\\!\\!\\!\int_{M}m_{\varepsilon}\log(m_{\varepsilon})\,dxdt=O(\varepsilon|\log\varepsilon|)$

from estimate (3.9). This yields (6.8).

Finally, we observe that we can build a smooth approximation of the
Wasserstein geodesic if we approximate $m_{0},m_{1}$ with the heat semigroup,
namely $m_{0\varepsilon}=S_{\varepsilon}(m_{0})$,
$m_{1\varepsilon}=S_{\varepsilon}(m_{1})$, where $S_{t}$ is the heat semigroup
as in Proposition 2.2. By using Li-Yau estimates on the heat kernel, in the
improved form given e.g. in [29, Thm 1.8] for Riemannian manifolds with Ricci
curvature bounded below, we have that there exists a constant $C$, only
depending on $M,d$, such that

$\varepsilon\,\left(\frac{|\nabla
S_{\varepsilon}(m_{0})|}{S_{\varepsilon}(m_{0})}+|\log(S_{\varepsilon}(m_{0}))|\right)\leq
C\,.$

This means that $\varepsilon\log(m_{0\varepsilon})$ is bounded in
$W^{1,\infty}(M)$, and so is for $\varepsilon\log(m_{1\varepsilon})$. From
(5.7) in Lemma 5.2 we deduce that $u_{\varepsilon}$ is uniformly bounded, and
then from Theorem 4.3 $u_{\varepsilon}$ is bounded in Lipschitz norm.
Moreover, at fixed $\varepsilon$, $(u_{\varepsilon},m_{\varepsilon})$ are
smooth according to Theorem 5.4. Finally, by Ascoli-Arzelá’s theorem,
$u_{\varepsilon}$ is relatively compact in $C^{0}(\overline{Q}_{T})$ and
converges uniformly towards its limit $u$. It is a standard result that $u$ is
a viscosity solution of the Hamilton Jacobi equation
$\partial_{t}u=\frac{1}{2}|\nabla u|^{2}$. ∎

##  7 Appendix A: existence of smooth solutions

Here we show the existence of solutions to the differential system

$\begin{cases}-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}u\right)+\rho u+\varepsilon\nabla
u\operatorname{\cdot_{g}\\!}\nabla V(x)=0&\text{in $Q$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}=\delta
u+\varepsilon(\log(m_{1})+V(x))&\text{in $\Sigma_{T}$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}+\delta
u=\varepsilon(\log(m_{0})+V(x))&\text{in $\Sigma_{0}$}\\\ \end{cases}$ (7.1)

and we recall (see (3.16)) that the expanded form of the first equation is

$-\partial_{tt}u+2\nabla(\partial_{t}u)\operatorname{\cdot_{g}\\!}\nabla
u-\varepsilon\Delta_{g}u-(\nabla^{2}u)(\nabla u,\nabla u)+\rho
u+\varepsilon\nabla u\operatorname{\cdot_{g}\\!}\nabla V=0.$ (7.2)

######  Proposition 7.1:

Let $\rho,\delta,\varepsilon>0$. Assume that $V\in W^{2,\infty}(M)$ and
$m_{0},m_{1}\in{\mathcal{P}}(M)\cap C^{1,\alpha}(M)$ with $m_{0},m_{1}>0$ in
$M$.
Then there exists a classical solution $u\in C^{2,\alpha}(\overline{Q}_{T})$
of the quasilinear elliptic problem (7.1).

We will prove such result by means of the method of continuity. For
convenience of notation, we set $\varepsilon=1$ in what follows. We consider
the differential operators $F:C^{2,\alpha}(\overline{Q}_{T})\rightarrow
C^{\alpha}(\overline{Q}_{T})$ and $G:C^{2,\alpha}(\overline{Q}_{T})\rightarrow
C^{1,\alpha}(\Sigma_{0}\cup\Sigma_{T})$ defined by

$\displaystyle F[u]$ $\displaystyle\coloneqq-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}u\right)+\rho u+\nabla
u\operatorname{\cdot_{g}\\!}\nabla V(x)$ $\displaystyle G[u]$
$\displaystyle\coloneqq\begin{cases}-\partial_{t}u+\frac{1}{2}\left|\nabla
u\right|^{2}-\delta u-\log(m_{1})-V(x)&\text{in $\Sigma_{T}$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}+\delta
u-\log(m_{0})-V(x)&\text{in $\Sigma_{0}$.}\\\ \end{cases}$

Finally we define the operator

$\begin{array}[]{rcl}P:C^{2,\alpha}(\overline{Q}_{T})&\longrightarrow&C^{\alpha}(\overline{Q}_{T})\times
C^{1,\alpha}(\Sigma_{0}\cup\Sigma_{T})\\\
u&\longrightarrow&\left(F[u],G[u]\right)\end{array}$

and the set

$E\coloneqq\Set{u\in C^{2,\alpha}(\overline{Q}_{T})}{\exists\tau\in[0,1]\quad
s.t.\quad P[u]=(1-\tau)P[0]}$

We note that $0\in E$ and that if a function $u\in
C^{2,\alpha}(\overline{Q}_{T})$ satisfies $P[u]=0$ then it is a solution for
the elliptic problem (7.1).

######  Lemma 7.2:

The set $E$ is bounded in $C^{1,\alpha}(\overline{Q}_{T})$.

###### Proof.

In the expanded form, given $\tau\in[0,1]$, the problem $P[u]=(1-\tau)P[0]$ is

$\begin{cases}-tr\left(\mathcal{A}(x,\nabla
u)\circ\overline{\nabla}^{2}u\right)+\rho u+\nabla
u\operatorname{\cdot_{g}\\!}\nabla V(x)=0&\text{in $Q_{T}$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}=\delta
u+\tau\left(\log(m_{1})+V(x)\right)&\text{in $\Sigma_{T}$}\\\
-\partial_{t}u+\frac{1}{2}\left|\nabla u\right|^{2}+\delta
u=\tau\left(\log(m_{0})+V(x)\right)&\text{in $\Sigma_{0}$}\\\ \end{cases}$
(7.3)

So by Theorem 4.3, (4.5), and $\tau\leq 1$ we get

$\displaystyle\lVert u\rVert_{W^{1,\infty}}$ $\displaystyle\leq
C(\delta,\lVert\tau\log(m_{0})\rVert_{W^{1,\infty}},\lVert\tau\log(m_{1})\rVert_{W^{1,\infty}},\lVert\tau
V\rVert_{W^{2,\infty}})$ $\displaystyle\leq
C(\delta,\lVert\log(m_{0})\rVert_{W^{1,\infty}},\lVert\log(m_{1})\rVert_{W^{1,\infty}},\lVert
V\rVert_{W^{2,\infty}})\,.$

Moreover, since $V\in W^{2,\infty}$, we get that the coefficients of the
elliptic problem $P[u]=(1-\tau)P[0]$ are $C^{0,\alpha}$ in the interior and
$C^{1,\alpha}$ on the boundary with respect to $(t,x)$, independently of
$\tau$.
Fixed any local system of coordinates on $M$, we recall that the second
covariant derivatives of $\psi$ are

$\nabla_{ij}\psi=\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}\psi-\sum_{k}\Gamma_{ij}^{k}\frac{\partial}{\partial x_{k}}\psi$

where the $\Gamma_{ij}^{k}$ are the Christoffel symbols.

Hence, if we localize the differential system (7.3), we get a differential
problem on $\mathbb{R}^{d}$ which differs from (7.3) only in the first order
terms (because of the Christoffel symbols, which are smooth), so an elliptic
problem with Hölder coefficients in their arguments. Therefore, we can apply
the classical results on Schauder estimates (e.g. [18], and [32, Lemma 2.4]
for the boundary estimates) in every local chart of a finite atlas ($M$ is
compact). We obtain a global $C^{1,\alpha}(\overline{Q}_{T})$ estimate on $u$,
independent of $\tau$. ∎

We now observe that, thanks to Lemma 17.29 in [18], we have that $E$ is closed
in $C^{2,\alpha}(\overline{Q}_{T})$ and that the set

$S\coloneqq\Set{\tau\in[0,1]}{\exists u\in C^{2,\alpha}(\overline{Q}_{T})\quad
s.t.\quad P[u]=(1-\tau)P[0]}$

is closed. Note that $S$ is not empty because $0\in S$. We want to prove that
$S$ is also open, so that it will coincide with $[0,1]$. To this purpose, we
prove that for each $u\in E$ the linear differential system induced by the
Gateaux derivative of $P$ has one and only one solution.

######  Lemma 7.3:

Given $u\in E$, $\phi\in
C^{0,\alpha}(\overline{Q}_{T}),\zeta_{1},\zeta_{0}\in{C^{1,\alpha}(M)}$ there
exists one and only one solution $\psi\in C^{2,\alpha}(\overline{Q}_{T})$ of
the linear problem