# Uniqueness for Keller-Segel-type chemotaxis models

J. A. Carrillo, S. Lisini, E. Mainini  José A. Carrillo, Department of
Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ,
UK<EMAIL_ADDRESS>Stefano Lisini, Dipartimento di Matematica “F.
Casorati”, Università degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy.
<EMAIL_ADDRESS>Edoardo Mainini, Università degli Studi di Genova,
Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti
(DIME) - sezione MAT. P.le Kennedy 1, 16129 Genova, Italy
<EMAIL_ADDRESS>
###### Abstract.

We prove uniqueness in the class of integrable and bounded nonnegative
solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our
proof works for the fully parabolic KS model, it includes the classical
parabolic-elliptic KS equation as a particular case, and it can be generalized
to nonlinear diffusions in the particle density equation as long as the
diffusion satisfies the classical McCann displacement convexity condition. The
strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and
the above-the-tangent characterizations of displacement convexity. As a
consequence, the displacement convexity of the free energy functional
associated to the KS system is obtained from its evolution for bounded
integrable initial data.

††AMS 2010 subject classification: 35A15

##  1\. Introduction

The classical Keller-Segel (KS) model for chemotaxis is the system

$\left\\{\begin{array}[]{rl}\partial_{t}n&=\kappa\Delta
n-\chi\,\mathrm{div}\,(n\nabla c),\\\ \partial_{t}c&=\eta\Delta c+\theta
n-\gamma c.\end{array}\right.$

Here, $n$ is the number/mass density of a bacteria/cell population and $c$
represents the concentration of a chemical attractant that can suffer chemical
degradation and that it is produced by the cells themselves due to chemotactic
interaction. The parameters $\kappa,\chi,\eta,\theta,\gamma$ might be suitable
functions, assumed to be constant in this simplified model. We can perform a
time scaling and a suitable change of variables, that is $\tau=\kappa t$,
$\rho(x,\tau)=\frac{\theta\chi}{\eta\kappa}n(x,\tau/\kappa),v(x,\tau)=\frac{\chi}{\kappa}c(x,\tau/\kappa)$.
The system is therefore reduced to

$\left\\{\begin{array}[]{rl}\partial_{t}\rho&=\Delta\rho-\,\mathrm{div}\,(\rho\nabla
v),\\\\[5.69054pt] \varepsilon\partial_{t}v&=\Delta v+\rho-\alpha
v,\end{array}\right.$ (1.1)

where $\alpha\geq 0$ and $\varepsilon\geq 0$ are constants
($\alpha=\gamma/\eta,\>\varepsilon=\kappa/\eta$). In case $\varepsilon=0$, it
restricts to the classical parabolic-elliptic Patlak-KS model

$\left\\{\begin{array}[]{l}\partial_{t}\rho=\Delta\rho-\,\mathrm{div}\,(\rho\nabla
v),\\\\[5.69054pt] -\Delta v+\alpha v=\rho.\end{array}\right.$ (1.2)

For $\varepsilon>0$, the natural free energy functional associated to the
dynamics of the system (1.1) is

$\mathscr{F}_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\rho\log\rho-v\rho)\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx\,.$ (1.3)

In the case $\varepsilon=0$, corresponding to (1.2), this Liapunov functional
is at least formally equivalent to

$\mathscr{F}_{0,\alpha}(\rho):=\int_{\mathbb{R}^{d}}(\rho\log\rho-\frac{1}{2}v\rho)\,dx\,$
(1.4)

with the convention that $v$ is obtained from the density $\rho$ by
$v=\mathcal{B}_{\alpha,d}\ast\rho$. Here, $\mathcal{B}_{\alpha,d}$ denotes the
Bessel kernel for $\alpha>0$ or the Newtonian kernel for $\alpha=0$, for any
dimension $d$. Therefore the role of the parameter $\varepsilon$ is to
discriminate between parabolic-parabolic and parabolic-elliptic system. Note
that the Liapunov functionals (1.3) and (1.4) are just formally equivalent
since the $L^{2}$-integrability of $\nabla\mathcal{B}_{\alpha,d}\ast\rho$
fails if $d=1,2$ and $\alpha=0$. Thus, even if the classical free energy
writing and valid for all cases when $\varepsilon=0$ is the one in (1.4), we
will prefer to work with the functional as in (1.3) even if $\varepsilon=0$,
with a suitable renormalization for the cases $d=1,2$ and $\alpha=0$ discussed
in Section 3.

Our main objective is the uniqueness of certain solutions, for both systems
(1.1) and (1.2). Let us introduce the notion of solution for the Cauchy
problems associated to (1.1) and (1.2) that we will consider in this work. We
denote by $\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$ the set of
nonnegative densities over $\mathbb{R}^{d}$ with mass $\mathfrak{m}$ and
finite second moment, i.e.,

$\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m}):=\left\\{\rho\in
L^{1}(\mathbb{R}^{d}):\rho\geq
0,\int_{\mathbb{R}^{d}}\rho(x)\,dx=\mathfrak{m},\int_{\mathbb{R}^{d}}|x|^{2}\rho(x)\,dx<+\infty\right\\}.$

######  Definition 1.1.

We say that a weakly continuous map $\rho\in
C_{w}([0,T];\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m}))$ is a bounded
solution to the Cauchy problem for (1.2), with initial datum
$\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$, if

  * i)

$\rho\in L^{\infty}((0,T)\times\mathbb{R}^{d})$ and $|x|^{2}\rho_{t}(x)\in
L^{\infty}((0,T),L^{1}(\mathbb{R}^{d}))$,

  * ii)

$\rho_{0}=\rho^{0}$ and the first equation of (1.2) holds in the sense of
distributions on $(0,T)\times\mathbb{R}^{d}$, where
$v_{t}=\mathcal{B}_{\alpha,d}\ast\rho_{t}$ for all $t\in[0,T]$,

  * iii)

$\rho_{t}\in W^{1,1}(\mathbb{R}^{d})$ for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$
and

$\int_{0}^{T}\int_{\mathbb{R}^{d}}\frac{|\nabla\rho_{t}(x)|^{2}}{\rho_{t}(x)}\,dx\,dt<+\infty.$
(1.5)

######  Definition 1.2.

We say that the couple of functions $(\rho,v)$, with $\rho\in
C_{w}([0,T];\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m}))$ and $v\in
L^{2}((0,T);W^{1,\,2}(\mathbb{R}^{d}))$, is a bounded solution to (1.1) with
initial datum $(\rho^{0},v^{0})$,
$\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and $v^{0}\in W^{1,2}(\mathbb{R}^{d})$, if

  * I)

$\rho\in L^{\infty}((0,T)\times\mathbb{R}^{d})$ and $|x|^{2}\rho_{t}(x)\in
L^{\infty}((0,T),L^{1}(\mathbb{R}^{d}))$,

  * II)

$\rho_{0}=\rho^{0}$, the first equation of (1.1) holds in the sense of
distributions on $(0,T)\times\mathbb{R}^{d}$, and $v$ is the unique solution
to the Cauchy problem for the forced parabolic equation
$\varepsilon\partial_{t}v-\Delta v+\alpha v=\rho$ over
$(0,T)\times\mathbb{R}^{d}$ in the standard sense, with initial datum $v^{0}$,

  * III)

the property iii) of Definition 1.1 holds.

Let us emphasize that the main properties we need to get uniqueness of
solution are the boundedness of the densities and the Fisher information
(1.5). They together imply that the velocity field of the continuity equation
for the density $\rho$ is a well defined object belonging to the right
functional space, see section 2 for details. Moreover, the boundedness of the
density implies that we have a uniform in bounded time intervals estimate on
the quasi-Lipschitz constant of part of the velocity field. These are the
basic properties that imply the uniqueness for bounded solutions. Let us
finally mention that part of the strategy is related to the uniqueness of
solutions to fluid and aggregation equations developed in [34, 27, 5, 28, 19,
7, 29]. The main novelty here is the interplay between the diffusive and the
aggregation parts. The main results of this work are:

######  Theorem 1.3.

Let $T>0$ and let $\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$. Let $\rho_{1},\rho_{2}$ be two bounded solutions
on $[0,T]\times\mathbb{R}^{d}$ to the Cauchy problem associated to (1.2), with
initial datum $\rho^{0}$. Then $\rho_{1}=\rho_{2}$.

######  Theorem 1.4.

Let $T>0$ and let $\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$, $v^{0}\in W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d})$. Let $(\rho_{1},v_{1})$ and $(\rho_{2},v_{2})$
be two bounded solutions on $[0,T]\times\mathbb{R}^{d}$ of the Cauchy problem
associated to (1.1), with initial datum $(\rho^{0},v^{0})$. Then
$(\rho_{1},v_{1})=(\rho_{2},v_{2})$.

The proof of uniqueness as stated in Theorems 1.3 and 1.4 will be a
consequence of a more general property: we will show that bounded solutions
satisfies a strong gradient flow formulation by means of a family of evolution
variational inequalities. This formulation is similar to the one for semi-
convex functionals and implies a non-expansivity property of the distance
between two solutions. This non-expansivity property yields uniqueness. All
these results will be stated in Theorem 3.1. Moreover this formulation lead
also to a relaxed convexity property of the energy functional as stated in
Theorem 4.1.

There is a huge literature about the KS system and their variations, so we
just restrict here to discuss the main results concerning about bounded
solutions. In the classical parabolic-elliptic KS equation
$\varepsilon=\alpha=0$ and $d=2$, global in time bounded solutions in the
subcritical case $\mathfrak{m}<8\pi$ have been obtained joining the results in
[12, 24, 14]. Actually, the global existence of weak solutions satisfying all
properties in Definition 1.1 except the $L^{\infty}$ bound was obtained in
[12] while $L^{\infty}$-bounds in bounded time intervals can be obtained from
the results in [24, 14]. The same techniques could eventually be used to get
local in time bounded solutions for all masses, although such a result is not
present in the literature. Let us also mention the recent preprint [17] in
which the authors actually show that the $L^{\infty}$-norm of the solution
decays in time like for the heat equation in the subcritical case
$\mathfrak{m}<8\pi$ for more restricted initial data. $L^{\infty}$-apriori
estimates were obtained in the classical parabolic-elliptic KS equation
$\varepsilon=0$ with $d\geq 2$ and $\alpha\geq 0$ for small $L^{d/2}$ initial
data in [20, 21]. These results together with similar arguments as in [12] to
get the free energy dissipation property and thus the Fisher information
bounds, could lead to the existence of bounded solutions in these cases. We
emphasize that these $L^{\infty}$ estimates show that the solution in bounded
time intervals is bounded by a constant that depends only on the
$L^{\infty}$-norm of the initial data, the initial free energy, and the final
time. In particular, existence of bounded solutions is expected if
$\rho^{0}\in L^{\infty}(\mathbb{R}^{d})$, and this explains the presence of
such an assumption in the previous definitions.

Concerning the fully parabolic KS system, we find global in time solutions
satisfying all properties stated in Definition 1.2 except the $L^{\infty}$
bounds in [15] for $d=2$ and the subcritical mass case $\mathfrak{m}<8\pi$.
$L^{\infty}$-apriori estimates were obtained in [25] for the fully parabolic
case but in bounded domains. It is reasonable to expect that this strategy
should work for the whole space case, although it is not written as such in
the literature. Results in higher dimensions concerning solutions with
$L^{\infty}$ estimates for small initial data can be found in [8] but
estimates on the free energy dissipation are missing there. We finally refer
to [11, 13, 14, 22] for different results concerning the existence of
solutions satisfying the boundedness of the Fisher information and/or the
uniform bounds of the solutions for particular choices of $\varepsilon\geq 0$,
$\alpha\geq 0$, and nonlinear diffusions.

As mentioned before, Theorems 1.3 and 1.4 are based on the derivation of
quasi-Lipschitz estimates for the chemoattractant $v$. This is the reason
behind the additional assumption on the initial datum $v^{0}$ in Theorem 1.4.
We will clarify the use of quasi-Lipschitz estimates of the chemoattractant in
the next section together with a quick summary of the main properties of
optimal transport that we need in this work. Section 3 is devoted to show the
main uniqueness results, derived from a more general property of bounded
solutions for the Keller-Segel model. In fact, we will show that for bounded
solutions we can obtain evolution variational inequalities. In Section 4 we
show that these evolution variational inequalities lead to certain convexity
of the associated free energy functional. In order not to break the flow of
the argument, we postpone to Section 5 the rigorous derivation of the quasi-
Lipschitz estimates of the elliptic and parabolic equations for $v$. In
Section 5, we will also prove a strengthening of Theorem 1.4, with more
general initial data. Finally, Section 6 is devoted to show how to adapt these
arguments to Keller-Segel models with nonlinear diffusion.

##  2\. Preliminary notions

###  2.1. Some elliptic and parabolic regularity estimates

The proofs of our results are based on the technique used by Yudovich [34] for
treating uniqueness in the case of incompressible Euler equations for
fluidodynamics. In particular, we exploit a quasi-Lipschitz property for the
velocity field of the continuity equation for $\rho$ in (1.1) and (1.2). This
property comes from the regularity that $v$ gains being solution to the second
equation in (1.1) and (1.2).

Suppose first that $v=\mathcal{B}_{0,d}\ast\rho$. If $\rho\in L^{1}\cap
L^{\infty}(\mathbb{R}^{d})$, by exploiting some estimates of the Newtonian
potential, $\nabla v$ satisfies the following log-Lipschitz property (see [6]
and [30, Chapter 8], [32] and also [34]),

$|\nabla v(x)-\nabla v(y)|\leq C|x-y|(1-\log^{-}|x-y|),$

where $C$ is a suitable positive constant, depending only on
$\|\rho\|_{L^{1}}$ and $\|\rho\|_{L^{\infty}}$ and $\log^{-}$ denotes the
negative part of the natural logarithm function. As a consequence, we get the
estimate

$|\nabla v(x)-\nabla v(y)|^{2}\leq C^{2}\varphi(|x-y|^{2})$ (2.1)

for some new positive constant $C$, where $\varphi$ is the concave function on
$[0,\infty)$ defined as

$\varphi(x):=\left\\{\begin{array}[]{ll}x\log^{2}x&\mbox{if $x\leq
e^{-1-\sqrt{2}}$},\\\ x+2(1+\sqrt{2})e^{-1-\sqrt{2}}&\mbox{if
$x>e^{-1-\sqrt{2}}$}.\end{array}\right.$ (2.2)

Indeed, for large values of $|x-y|$ the estimate (2.1) is quite obvious, since
it is immediate to show that $\nabla\mathcal{B}_{0,d}\ast\rho$ is a bounded
function in the whole space with a direct estimate using the fact that
$\rho\in L^{1}\cap L^{\infty}(\mathbb{R}^{d})$. The log-Lipschitz property
itself can be justified through standard elliptic regularity, as we will do in
Section 5.

Analogous facts hold if we consider the equation $-\Delta v+\alpha v=\rho$,
appearing in (1.2), or more general uniformly elliptic operators, so that we
have

######  Proposition 2.1.

Suppose that $\rho\in L^{1}\cap L^{\infty}({\mathbb{R}^{d}})$ and $\alpha\geq
0$. Then $v=\mathcal{B}_{\alpha,d}\ast\rho$ satisfies the estimate (2.1),
where $C$ is a suitable positive constant, depending only on
$\alpha,d,\|\rho\|_{L^{1}(\mathbb{R}^{d})}$, and
$\|\rho\|_{L^{\infty}(\mathbb{R}^{d})}$.

About the parabolic equation for $v$ in (1.1), the quasi-Lipschitz property
also carries over, since formally inequality (2.1) translates in terms of the
parabolic metric to

$|\nabla v(t,x)-\nabla v(s,y)|^{2}\leq
C^{2}\varphi((|x-y|+|s-t|^{1/2})^{2})\quad\forall
x,y\in\mathbb{R}^{d},\;s,t\in[0,T].$ (2.3)

######  Proposition 2.2.

Suppose that $\rho\in L^{\infty}((0,T)\times{\mathbb{R}^{d}})$, $v^{0}\in
W^{2,\infty}(\mathbb{R}^{d})$ and $\alpha\geq 0$. If $v$ is the unique
solution to the Cauchy problem for the parabolic equation
$\partial_{t}v=\Delta v-\alpha v+\rho$ (in the standard sense of convolution
with fundamental solution), then $v$ satisfies (2.3), where $C$ is a suitable
positive constant, depending only on
$\alpha,d,\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$, and
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$.

For a more complete insight into these properties, it will be convenient to
recall some facts about the Zygmund class and its role in elliptic and
parabolic regularity. However, in order not to introduce some not really
necessary notation before the proof of our main results, we prefer to postpone
the proof of Proposition 2.1 and Proposition 2.2 to Section 5. Indeed, in
Section 5 we will develop a more rigorous discussion about the log-Lipschitz
estimates, and thanks to some refined parabolic regularity we will also prove
a slight strengthening of Theorem 1.4.

###  2.2. Elementary notions of optimal transport

Given $\rho_{0},\rho_{1}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$, we
define the Wasserstein distance between $\rho_{0}$ and $\rho_{1}$ as

$W_{2}(\rho_{0},\rho_{1})=\left(\int_{\mathbb{R}^{d}}|x-\mathcal{T}(x)|^{2}\,\rho_{0}(x)\,dx\right)^{\frac{1}{2}},$

where $\mathcal{T}$ is the unique optimal transport map between $\rho_{0}$ and
$\rho_{1}$, that is, the map $\mathcal{T}:\mathbb{R}^{d}\to\mathbb{R}^{d}$
which minimizes $\int_{\mathbb{R}^{d}}|x-\mathcal{S}(x)|^{2}\,\rho_{0}(x)\,dx$
among all the Borel maps $\mathcal{S}:\mathbb{R}^{d}\to\mathbb{R}^{d}$
satisfying $\mathcal{S}_{\\#}\rho_{0}=\rho_{1}$. We recall that
$\mathcal{S}_{\\#}\rho_{0}=\rho_{1}$ means that
$\int_{\mathbb{R}^{d}}\varphi(x)\rho_{1}(x)\,dx=\int_{\mathbb{R}^{d}}\varphi(\mathcal{S}(x))\rho_{0}(x)\,dx$
for every continuous and bounded function
$\varphi:\mathbb{R}^{d}\to\mathbb{R}^{d}$.

The Wasserstein geodesic between $\rho_{0}$ and $\rho_{1}$ is the curve
$s\in[0,1]\mapsto\rho^{s}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$
defined by the so-called displacement interpolation along the optimal
transport map $\mathcal{T}$ between $\rho_{0}$ and $\rho_{1}$, that is,
$\rho^{s}:=((1-s)\mbox{\boldmath$\mathfrak{i}$}+s\mathcal{T})_{\\#}\rho_{0}$.
In particular, for any $s$,
$\mathcal{T}_{s}:=(1-s)\mbox{\boldmath$\mathfrak{i}$}+s\mathcal{T}$ is the
optimal map between $\rho_{0}$ and $\rho^{s}$ and there holds
$W_{2}(\rho^{r},\rho^{s})=|s-r|W_{2}(\rho_{0},\rho_{1})$.

We recall a formula for the differentiation of the squared Wasserstein
distance along solutions of the continuity equation. Let
$t\in[0,T]\mapsto\rho_{t}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$ be a
weakly continuous curve which is distributional solution of

$\partial_{t}\rho_{t}+\mathrm{div}\,(\xi_{t}\rho_{t})=0,$

for some Borel velocity field $\xi_{t}$ such that
$\int_{0}^{T}\|\xi_{t}\|^{2}_{L^{2}(\mathbb{R}^{d},\rho_{t};\mathbb{R}^{d})}\,dt<+\infty$.
Then the curve is absolutely continuous with respect to the Wasserstein
distance, [3, Theorem 8.3.1]. Then, for any
$\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$, it holds

$\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})=\int_{\mathbb{R}^{d}}\langle\xi_{t}(x),x-\mathcal{T}_{t}(x)\rangle\,\rho_{t}(x)\,dx,\qquad\mbox{for
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$},$ (2.4)

where $\mathcal{T}_{t}$ is the optimal map between $\rho_{t}$ and $\bar{\rho}$
(see [3, Theorem 8.4.7, Remark 8.4.8]).

Finally, let us recall an estimate relating the $2$-Wasserstein distance and
the $H^{-1}$ norm proved in [27, Proposition 2.8]. Given two nonnegative
densities with the same mass
$\rho_{1},\rho_{2}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$, there holds

$\|\rho_{1}-\rho_{2}\|_{\dot{H}^{-1}(\mathbb{R}^{d})}\leq\max\\{\|\rho_{1}\|_{\infty},\|\rho_{2}\|_{\infty}\\}^{1/2}W_{2}(\rho_{1},\rho_{2}).$
(2.5)

Here we are letting $\dot{H}^{1}(\mathbb{R}^{d})$ be the space of Lebesgue
measurable functions $v:\mathbb{R}^{d}\to\mathbb{R}$ such that $\|\nabla
v\|_{L^{2}(\mathbb{R}^{d})}<+\infty$, so that $\dot{H}^{-1}(\mathbb{R}^{d})$
is defined by duality with functions having finite $L^{2}(\mathbb{R}^{d})$
norm of the gradient only. By the way, we can also consider the space
$H^{1}(\mathbb{R}^{d})=W^{1,2}(\mathbb{R}^{d})$. In fact, from the proof in
[27, Proposition 2.8] it is not difficult to see that the same estimate holds
considering the $H^{-1}(\mathbb{R}^{d})$ space given by duality with the full
norm $(\|\nabla
v\|^{2}_{L^{2}(\mathbb{R}^{d})}+\|v\|^{2}_{L^{2}(\mathbb{R}^{d})})^{1/2}$.

##  3\. Bounded solutions as gradient flows: EVI and uniqueness

The uniqueness Theorems 1.3 and 1.4 are consequences of a general result
interpreting bounded solutions to (1.1) (resp. (1.2)) as the trajectory of the
gradient flow of the functional (1.3) (resp. (1.4)) in the appropriate metric
setting. We prove that bounded solutions satisfy a family of evolution
variational inequalities (EVI). Among different notions of gradient flow in
metric sense, the EVI formulation is stronger than other formulations and
typically corresponding to a convex structure, as in [3, Theorem 11.2.1] for
the theory in the Wasserstein setting.

Notation for the energy functional. Before giving the proof, we introduce some
uniform notation for working with the full functional (1.3) even in the
parabolic-elliptic case. Let
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$. We are considering the free energy functional

$\mathscr{F}_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\rho\log\rho-v\rho)\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx\,,$

defined for $v$ being any $W^{1,2}(\mathbb{R}^{d})$ function if
$\varepsilon>0$. On the other hand, if $\varepsilon=0$ it is understood that
$v$ is given by $\mathcal{B}_{\alpha,d}\ast\rho$. Therefore the parameter
$\varepsilon$ only indicates if we are considering problem (1.1) or (1.2). In
particular, this writing of the functional as in (1.3) is valid in general,
even for $\varepsilon=0$, except for two particular cases:
$\varepsilon=\alpha=0$ and $d=1,2$, as discussed in the introduction. In these
two cases, we need to renormalize the free energy functional. Given
$\rho^{*}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$ a smooth and
compactly supported density and $v^{*}=\mathcal{B}_{0,d}\ast\rho^{*}$, we
redefine (1.3) for $\varepsilon=\alpha=0$ and $d=1,2$ as

$\mathscr{F}_{0,0}(\rho,v):=\int_{\mathbb{R}^{d}}\left[\rho\log\rho-v(\rho-\rho^{*})\right]\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}|\nabla(v-v^{*})|^{2}\,dx-\frac{1}{2}\int_{\mathbb{R}^{d}}\rho^{*}v^{*}\,dx\,.$
(3.1)

Notice that $\nabla(v-v^{*})\in L^{2}(\mathbb{R}^{d})$, as $\rho-\rho^{*}$ has
zero mean, see [4, 32] for more details.

In the rest of this work, when referring to the free energy functional
$\mathscr{F}_{\varepsilon,\alpha}(\rho,v)$, we will be using (1.3) for any
$\varepsilon\geq 0,\alpha\geq 0$, except for $\varepsilon=\alpha=0$ and
$d=1,2$ where the free energy functional is given by (3.1).

Let us observe that now all the integrals involved in the definition of
$\mathscr{F}_{\varepsilon,\alpha}$ are well defined and finite for
$\varepsilon\geq 0,\alpha\geq 0$ and $\rho,v$ as above. The negative part of
the entropy term can be classically treated by the Carleman inequality, see
for instance [9, Lemma 2.2] where the second moment bound on the density is
used. The boundedness of the density controls the positive contribution of the
entropy term together with the integrability of $v\rho$ in case
$\varepsilon>0$ since $v\in W^{1,2}(\mathbb{R}^{d})$. For $\varepsilon=0$ the
integrability of $v\rho$ in case $\alpha>0$ is implied by the Newtonian
potential case $\alpha=0$ since the singularity of the Bessel potential at the
origin is the same. The integrability for $\alpha=\varepsilon=0$ and $d\geq 3$
results directly from the Hardy-Littlewood-Sobolev inequality for the
Newtonian potential. For $\alpha=\varepsilon=0$ and $d=1,2$ we use the
behavior at infinity of the density $\rho$. Actually, $\alpha=\varepsilon=0$
and $d=1$ is a trivial case since the Newtonian potential is given by
$\mathcal{B}_{0,1}(x)=|x|$. For $\alpha=\varepsilon=0$ and $d=2$ since
$\log(e+|x|^{2})\rho\in L^{1}(\mathbb{R}^{d})$ then $v\rho\in
L^{1}(\mathbb{R}^{d})$ using the logarithmic HLS inequality, see for instance
[10].

Notation for the ambient metric space. We let
$X_{\varepsilon}:=\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\times
L^{2}(\mathbb{R}^{d})$ endowed with the distance

$D^{2}(z_{1},z_{2})=D^{2}((\rho_{1},v_{1}),(\rho_{2},v_{2}))=W_{2}^{2}(\rho_{1},\rho_{2})+\varepsilon\|v_{1}-v_{2}\|^{2}_{L^{2}(\mathbb{R}^{d})}\,,$

with the convention that $X_{0}=\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$
and $D_{0}(z_{1},z_{2})=W_{2}(\rho_{1},\rho_{2})$. Moreover, for $z=\rho\in
X_{0}\times L^{\infty}({\mathbb{R}^{d}})$, $\mathscr{F}_{0,\alpha}(z)$ will be
understood to be $\mathscr{F}_{0,\alpha}(\rho,v)$ with
$v=\mathcal{B}_{\alpha,d}\ast\rho$, as usual when $\varepsilon=0$.

In the space $X_{\varepsilon}$ the metric derivative of an absolutely
continuous curve $t\mapsto z_{t}$ is denoted and defined by

$|z^{\prime}|_{D}(t)=\lim_{h\to 0}\frac{D(z_{t+h},z_{t})}{h},$

and it exists for $\mathcal{L}^{1}$-a.e. $t>0$. The local metric slope of the
functional $\mathscr{F}_{\varepsilon,\alpha}$ is defined by

$|\partial\mathscr{F}_{\varepsilon,\alpha}|_{D}(z):=\limsup_{D(\zeta,z)\to
0}\frac{(\mathscr{F}_{\varepsilon,\alpha}(z)-\mathscr{F}_{\varepsilon,\alpha}(\zeta))^{+}}{D(\zeta,z)}.$

These two abstractly defined objects are used to give the notion of curves of
maximal slope in general metric setting, see [2, §3], [3, Chapter 1]. The main
consequences of this gradient flow structure are summarized in the following
result.

Before stating the Theorem we define the function
$\omega:[0,+\infty)\to[0,+\infty)$ by

$\omega(x)=\sqrt{\mathfrak{m}x\varphi(\mathfrak{m}^{-1}x)},$ (3.2)

where $\varphi$ is defined in (2.2). Moreover, given a fixed $s_{0}>0$, we
define a strictly monotone continuous function
$G:[0,+\infty)\to[-\infty,+\infty)$ by
${G(s):=\int_{s_{0}}^{s}\frac{1}{\omega(r)}\,dr}$ for $s>0$ and $G(0)=-\infty$
(we observe that $G^{-1}:[-\infty,+\infty)\to[0,+\infty)$ is surjective).

######  Theorem 3.1.

Let $t\mapsto z_{t}=(\rho_{t},v_{t})$ be a bounded solution of problem (1.1)
for $\varepsilon>0$, starting from $z^{0}=(\rho^{0},v^{0})\in
X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$, according to Definition 1.2. If
$\varepsilon=0$, let $z_{t}=\rho_{t}$ be a bounded solution to problem (1.2),
starting from $z^{0}=\rho^{0}\in X_{0}\cap L^{\infty}(\mathbb{R}^{d})$,
according to Definition 1.1. Then the three following properties hold:

  * i)

The evolution variational inequality (EVI) formulation: for any
$\bar{z}=(\bar{\rho},\bar{v})\in
X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times
W^{1,2}(\mathbb{R}^{d})\big{)}$ (reduced to $\bar{z}=\bar{\rho}\in X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$), the map $t\mapsto
D^{2}(z_{t},\bar{z})$ is absolutely continuous and there exists a constant $C$
depending on $\|\rho\|_{L^{\infty}((0,T)\times\mathbb{R}^{d})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$ and
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$, such that

$\frac{1}{2}\frac{d}{dt}D^{2}(z_{t},\bar{z})\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{z})-\mathscr{F}_{\varepsilon,\alpha}(z_{t})+C\omega(D^{2}(z_{t},\bar{z}))\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$ (3.3)

  * ii)

The energy dissipation equality (EDE) in metric sense: the map
$t\mapsto\mathscr{F}_{\varepsilon,\alpha}(z_{t})$ is locally Lipschitz
continuous and

$\frac{d}{dt}{\mathscr{F}_{\varepsilon,\alpha}}(z_{t})=-\frac{1}{2}|\partial{\mathscr{F}_{\varepsilon,\alpha}}|_{D}^{2}(z_{t})-\frac{1}{2}|z^{\prime}|_{D}^{2}(t)\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$ (3.4)

  * iii)

The following expansion control property: given another bounded solution
$t\mapsto\zeta_{t}$, with initial datum $\zeta^{0}$ in the same space of
$z^{0}$ above, there exists a constant $C$, depending on
$\|\rho\|_{L^{\infty}((0,T)\times\mathbb{R}^{d})}$ and
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$ (and the same quantities associated
to $\zeta$), such that there holds

$D^{2}(z_{t},\zeta_{t})\leq
G^{-1}(G(D^{2}(z^{0},\zeta^{0}))+4Ct)\quad\text{for every }t\in[0,T).$ (3.5)

###### Proof.

We first introduce the auxiliary functional

$\Phi_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\rho\log\rho-v\rho)\,dx,$

for $\rho$ and $v$ being as in the definition of
$\mathscr{F}_{\varepsilon,\alpha}$ at the beginning of this section, so that

$\mathscr{F}_{\varepsilon,\alpha}(\rho,v)=\Phi_{\varepsilon,\alpha}(\rho,v)+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx\,$

and

$\Phi_{0,0}(\rho,v)=\mathscr{F}_{0,0}(\rho,v)-\frac{1}{2}\int_{\mathbb{R}^{d}}|\nabla(v-v^{*})|^{2}\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}\rho^{*}v^{*}\,dx-\int_{\mathbb{R}^{d}}\\!\rho^{*}v\,dx\,\quad\mbox{for
$d=1,2$.}$ (3.6)

The proof is organized in four steps.

Step1. Quasi-Lipschitz Estimate implies control of the evolution of the
Wasserstein distance.- Thanks to the assumption (1.5), we learn that the
Fisher information
${\int_{\mathbb{R}^{d}}\frac{|\nabla\rho_{t}(x)|^{2}}{\rho_{t}(x)}\,dx}$ is
finite for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$. Let
$\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$. Exploiting the differentiability properties of
the entropy functional, we can use the above-the-tangent formulation of
displacement convexity to get for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$

$\int_{\mathbb{R}^{d}}\bar{\rho}(x)\log\bar{\rho}(x)\,dx-\int_{\mathbb{R}^{d}}\rho_{t}(x)\log\rho_{t}(x)\,dx\geq\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx,$
(3.7)

where $\mathcal{T}_{t}$ denotes the optimal transport map between $\rho_{t}$
and $\bar{\rho}$. We refer to [2, §3.3.1] for an intuitive proof of this fact,
and to [3, Chapter 10] for the theory in full generality. In particular, the
finiteness of the Fisher information of $\rho_{t}$ implies that the second
term is finite, so that this differentiation formula is meaningful. If
$\varepsilon>0$ (resp. $\varepsilon=0$), let $\bar{v}\in
W^{1,2}(\mathbb{R}^{d})$ (resp.
$\bar{v}=\mathcal{B}_{\alpha,d}\ast\bar{\rho}$). Take

$I_{t}:=\Phi_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\Phi_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}(\bar{v}(x)-v_{t}(x))\bar{\rho}(x)\,dx\,.$

Using the notation $x^{s}_{t}:=(1-s)x+s\mathcal{T}_{t}(x)$, $s\in[0,1]$, and
taking into account that

$\displaystyle\int_{\mathbb{R}^{d}}v_{t}(x)(\bar{\rho}(x)-\rho_{t}(x))\,dx$
$\displaystyle=\int_{\mathbb{R}^{d}}(v_{t}(\mathcal{T}_{t}(x))-v_{t}(x))\rho_{t}(x)\,dx$
$\displaystyle=\int_{\mathbb{R}^{d}}(v_{t}(x_{t}^{1})-v_{t}(x_{t}^{0}))\rho_{t}(x)\,dx=\int_{0}^{1}\frac{d}{ds}\int_{\mathbb{R}^{d}}v_{t}(x_{t}^{s})\rho_{t}(x)\,dx\,ds$

and (3.7), we obtain for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$

$\displaystyle I_{t}$
$\displaystyle\geq\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-\int_{\mathbb{R}^{d}}v_{t}(x)(\bar{\rho}(x)-\rho_{t}(x))\,dx$
$\displaystyle=\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-\int_{0}^{1}\int_{\mathbb{R}^{d}}\langle\nabla
v_{t}(x^{s}_{t}),\mathcal{T}_{t}(x)-x\rangle\,\rho_{t}(x)\,dx\,ds$
$\displaystyle=\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x)-\rho_{t}(x)\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-\int_{0}^{1}\int_{\mathbb{R}^{d}}\langle\nabla
v_{t}(x^{s}_{t})-\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,\rho_{t}(x)\,dx\,ds.$

Let us denote by $II_{t}$ the last term in the right hand side above. The
crucial point is to treat such term using the log-Lipschitz property of
$\nabla v$. Notice that, if $\varepsilon=0$, we are in the assumptions of
Proposition 2.1 and we apply (2.1), where the constant $C$ depends in
principle only on ($\mathfrak{m},\alpha,d$ and) the $L^{\infty}$ norm of
$\rho_{t}$, which we are assuming to be uniformly bounded on $(0,T)$. In the
case $\varepsilon>0$, still by the uniform space-time $L^{\infty}$ assumption
on $\rho_{t}$ and the $W^{2,\infty}$ assumption on $v^{0}$, we are in the
framework of Proposition 2.2, so that we can apply the estimate (2.3). In this
case the constant will depend also on ($\varepsilon$ and)
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$. Since $\varphi$ is concave, we can
also use the Jensen inequality, and letting
$\rho^{s}_{t}={x^{s}_{t}}_{\\#}\rho_{t}$ be the Wasserstein geodesic
connecting $\rho_{t}$ and $\bar{\rho}$ we have

$\displaystyle\left|II_{t}\right|$ $\displaystyle\leq
W_{2}(\rho_{t},\bar{\rho})\int_{0}^{1}\left(\int_{\mathbb{R}^{d}}|\nabla
v_{t}(x^{s}_{t})-\nabla v_{t}(x)|^{2}\rho_{t}(x)\,dx\right)^{1/2}\,ds$ (3.8)
$\displaystyle\leq
CW_{2}(\rho_{t},\bar{\rho})\int_{0}^{1}\left(\int_{\mathbb{R}^{d}}\varphi(|x^{s}_{t}-x|^{2})\rho_{t}(x)\,dx\right)^{1/2}\,ds$
$\displaystyle\leq\sqrt{\mathfrak{m}}CW_{2}(\rho_{t},\bar{\rho})\int_{0}^{1}\sqrt{\varphi(\mathfrak{m}^{-1}W_{2}^{2}(\rho_{t},\rho^{s}_{t}))}\,ds\leq\sqrt{\mathfrak{m}}CW_{2}(\rho_{t},\bar{\rho})\sqrt{\varphi(\mathfrak{m}^{-1}W_{2}^{2}(\rho_{t},\bar{\rho}))}\,.$

The last inequality holds since geodesic interpolation ensures

$\int_{\mathbb{R}^{d}}|x-x^{s}_{t}|^{2}\rho_{t}(x)\,dx=W_{2}^{2}(\rho_{t},\rho^{s}_{t})=s^{2}W_{2}^{2}(\rho_{t},\bar{\rho})$

for all $s\in[0,1]$ and since $\varphi$ is non decreasing. We recall that the
constant $C$ in (3.8) depends only on ($\varepsilon$, $\alpha,d$, the mass
$\mathfrak{m}$ and) the $L^{\infty}((0,T)\times\mathbb{R}^{d})$ norm of $\rho$
and, in the case $\varepsilon>0$, the $W^{2,\infty}(\mathbb{R}^{d})$ norm of
$v^{0}$. Inserting this in the estimate for $I_{t}$, we have for
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$

$I_{t}\geq\int_{\mathbb{R}^{d}}\langle{\nabla\rho_{t}}(x)-\rho_{t}(x)\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx-C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))\,,$
(3.9)

where $\omega$ is the function defined in (3.2). Since $\rho_{t}$ satisfies
the continuity equation

$\partial_{t}\rho_{t}+\mathrm{div}\,(\xi_{t}\rho_{t})=0\qquad\mbox{with
}\quad\rho_{t}\xi_{t}=-{\nabla\rho_{t}}+\rho_{t}\nabla v_{t}$

and (1.5), the uniform $L^{\infty}$ bound of $\rho_{t}$ implies that
$\int_{0}^{T}\|\xi_{t}\|^{2}_{L^{2}(\mathbb{R}^{d},\rho_{t};\mathbb{R}^{d})}\,dt<+\infty$.
Therefore $t\mapsto\rho_{t}$ is absolutely continuous with respect to $W_{2}$
and by (2.4)

$\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})=\int_{\mathbb{R}^{d}}\langle\nabla\rho_{t}(x)-\rho_{t}(x)\nabla
v_{t}(x),\mathcal{T}_{t}(x)-x\rangle\,dx\qquad\mbox{for $\mathcal{L}^{1}$-a.e.
t}\in(0,T).$

Inserting this into (3.9), and recalling the definition of $I_{t}$, we finally
obtain

$\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})\leq\Phi_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\Phi_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}(\bar{v}-v_{t})\bar{\rho}\,dx+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))$
(3.10)

for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$.

Step 2: EVI for the parabolic-parabolic case.- Recalling that $\bar{v}\in
W^{1,\,2}(\mathbb{R}^{d})$, observing that $\Delta v_{t}\in
L^{2}(\mathbb{R}^{d})$ for a.e.-$t\in(0,T)$ and using the elementary identity
$|a|^{2}-|b|^{2}=|a-b|^{2}+2\langle b,a-b\rangle$ for every
$a,b\in\mathbb{R}^{k}$, the variation of the second part of the functional
(1.3) (that is, $\mathscr{F}_{\varepsilon,\alpha}-\Phi_{\varepsilon,\alpha}$)
can be written as

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{d}}\left[|\nabla\bar{v}|^{2}-\right.$
$\displaystyle\left.|\nabla
v_{t}|^{2}+\alpha(\bar{v}^{2}-v_{t}^{2})\right]\,dx$ (3.11)
$\displaystyle=\int_{\mathbb{R}^{d}}(\alpha v_{t}-\Delta
v_{t})(\bar{v}-v_{t})\,dx+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}$
$\displaystyle=\int_{\mathbb{R}^{d}}(\rho_{t}-\varepsilon\partial_{t}v_{t})(\bar{v}-v_{t})\,dx+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}$
$\displaystyle=\int_{\mathbb{R}^{d}}\rho_{t}(\bar{v}-v_{t})\,dx+\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}.$

Therefore, we deduce

$\displaystyle\mathscr{F}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\mathscr{F}_{\varepsilon,\alpha}(\rho_{t},v_{t})$
$\displaystyle=\Phi_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\Phi_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}\rho_{t}(\bar{v}-v_{t})\,dx$
(3.12)
$\displaystyle\quad+\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}+\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}.$

Now, we use again (3.10), leading to

$\displaystyle\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}$
$\displaystyle+\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\mathscr{F}_{\varepsilon,\alpha}(\rho_{t},v_{t})+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))$
(3.13)
$\displaystyle\quad+\int_{\mathbb{R}^{d}}(\bar{\rho}-\rho_{t})(\bar{v}-v_{t})\,dx-\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})}-\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}.$

By using the duality between $\dot{H}^{1}$ and $\dot{H}^{-1}$, the Young
inequality, and (2.5) we have

$\displaystyle\int_{\mathbb{R}^{d}}(\bar{\rho}-\rho_{t})(\bar{v}-v_{t})\,dx$
$\displaystyle\leq\|\bar{\rho}-\rho_{t}\|_{\dot{H}^{-1}(\mathbb{R}^{d})}\|\bar{v}-v_{t}\|_{\dot{H}^{1}(\mathbb{R}^{d})}\leq\frac{1}{2}\|\bar{\rho}-\rho_{t}\|^{2}_{\dot{H}^{-1}(\mathbb{R}^{d})}+\frac{1}{2}\|\nabla(\bar{v}-v_{t})\|^{2}_{L^{2}(\mathbb{R}^{d})}$
(3.14)
$\displaystyle\leq\frac{1}{2}QW_{2}^{2}(\bar{\rho},\rho_{t})+\frac{1}{2}\|\nabla(\bar{v}-v_{t})\|^{2}_{L^{2}(\mathbb{R}^{d})},$

where $Q$ is the largest of the $L^{\infty}$ norms of $\bar{\rho}$ and
$\rho_{t}$ over the time interval $(0,T)$. Taking into account that $\omega$
is given by (3.2) and that
$\sqrt{\mathfrak{m}\varphi(\mathfrak{m}^{-1}x^{2})}\geq x$ for every $x>0$,
combining (3.13) and (3.14) we get, up to introducing a new constant $C$,

$\displaystyle\frac{\varepsilon}{2}\frac{d}{dt}\|v_{t}-\bar{v}\|_{L^{2}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})$
$\displaystyle\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})-\mathscr{F}_{\varepsilon,\alpha}(\rho_{t},v_{t})+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))-\frac{\alpha}{2}\|v_{t}-\bar{v}\|^{2}_{L^{2}({\mathbb{R}^{d}})}$
(3.15)

for a.e. $t\in(0,T)$. The new constant $C$ depends as usual on ($\varepsilon$
$\alpha$, $d$, $\mathfrak{m}$ and)
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$,
$\|v^{0}\|_{W^{2,\infty}(\mathbb{R}^{d})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$.

Step 3: EVI for the parabolic-elliptic case.- When either $d\geq 3$ or
$\alpha>0$, we can repeat the proof of the parabolic-parabolic case, letting
$\varepsilon=0$ therein and recalling that $\bar{v}$ is no more an arbitrary
$W^{1,2}(\mathbb{R}^{d})$ function but is given by convolution with
$\bar{\rho}$. In particular we arrive to the corresponding of (3.13), and the
second line therein can now be estimated as follows. Using the inequality
$\|v\|_{H^{1}_{\alpha}(\mathbb{R}^{d})}\leq\|\rho\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}$
for $-\Delta v+\alpha v=\rho$, $\alpha>0$, where the notation is
$\|v\|^{2}_{H^{1}_{\alpha}(\mathbb{R}^{d})}:=\|\nabla
v\|^{2}_{L^{2}(\mathbb{R}^{d})}+\alpha\|v\|^{2}_{L^{2}(\mathbb{R}^{d})}$ (and
using $\dot{H}^{1}$ if $\alpha=0$), we get

$\int_{\mathbb{R}^{d}}(\bar{v}-v_{t})(\bar{\rho}-\rho_{t})\,dx\\!\leq\|\bar{v}-v_{t}\|_{H^{1}_{\alpha}(\mathbb{R}^{d})}\|\bar{\rho}-\rho_{t}\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}\leq\frac{1}{2}\|\bar{v}-v_{t}\|_{H^{1}_{\alpha}(\mathbb{R}^{d})}^{2}+\frac{1}{2}\|\bar{\rho}-\rho_{t}\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}^{2}.$

Moreover, recalling the estimate (2.5) (which works both in $\dot{H}^{-1}$ and
$H^{-1}_{\alpha}$) we have

$\|\bar{\rho}-\rho_{t}\|_{H^{-1}_{\alpha}(\mathbb{R}^{d})}\leq
QW_{2}^{2}(\bar{\rho},\rho_{t}),$

for all $t\in[0,T]$, where $Q$ is the largest of the $L^{\infty}$ norms of
$\bar{\rho}$ and $\rho_{t}$ over the time interval $[0,T]$. Inserting these
estimates in (3.13) we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}W_{2}^{2}(\rho_{t},\bar{\rho})\leq\,\mathscr{F}_{0,\alpha}(\bar{\rho})-\mathscr{F}_{0,\alpha}(\rho_{t})+C\omega(W_{2}^{2}(\rho_{t},\bar{\rho})),$
(3.16)

for $\mathcal{L}^{1}$-a.e. $t\in(0,T)$, where the constant $C$ depends only on
$\varepsilon$, $\alpha$, $d$, $\mathfrak{m}$,
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$.

In the case $\alpha=0$, $d=1,2$, we have to consider the functional in (3.1).
By using the identity

$\frac{1}{2}\|\nabla(\bar{v}-v^{*})\|^{2}_{L^{2}(\mathbb{R}^{d})}-\frac{1}{2}\|\nabla(v_{t}-v^{*})\|^{2}_{L^{2}(\mathbb{R}^{d})}=\int_{\mathbb{R}^{d}}(\rho_{t}-\rho^{*})(\bar{v}-v_{t})\,dx+\frac{1}{2}\|\nabla(v_{t}-\bar{v})\|^{2}_{L^{2}(\mathbb{R}^{d})},$

with similar computations as in (3.11), this time considering
$\mathscr{F}_{0,0}(\rho,v)-\Phi_{0,0}(\rho,v)$ as obtained from (3.6), we can
still find (3.12) and conclude obtaining again (3.16).

Step 4: Conclusion.- We are ready to prove the three points in the statement
of the theorem. The proof of i) is a consequence of (3.15) for the case
$\varepsilon>0$, and (3.16) for the case $\varepsilon=0$, taking into account
that $\alpha\geq 0$ and that
$\omega(D^{2}(z_{t},\bar{z}))\geq\omega(W_{2}^{2}(\rho_{t},\bar{\rho}))$ being
$\omega$ increasing.

It is a standard fact that the gradient flow formulation in EVI sense implies
the one in EDE sense in (3.4). Indeed, the proof of  ii) follows from (3.3)
and (3.5) and can be exactly carried out as in [2, Proposition 3.6].

The proof of (3.5) still follows from (3.3). Indeed we can apply [3, Lemma
4.3.4] (see also the argument of [3, Theorem 11.1.4]) and obtain that for
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$

$\frac{1}{2}\frac{d}{ds}D^{2}(z_{s},\zeta_{s}){\Big{|}}_{s=t}\leq\,\frac{1}{2}\frac{d}{ds}D^{2}(z_{s},\zeta_{t}){\Big{|}}_{s=t}+\frac{1}{2}\frac{d}{ds}D^{2}(z_{t},\zeta_{s}){\Big{|}}_{s=t}\leq
2C\omega(D^{2}(z_{t},\zeta_{t})).$ (3.17)

Here, $C=\max\\{C_{1},C_{2}\\}$, where $C_{1}$ is the supremum on $s\in(0,T)$
of the constant in (3.3) for $z_{t}$ with $\bar{z}=\zeta_{s}$, which is finite
since $\zeta\in L^{\infty}((0,T)\times\mathbb{R}^{d})$, and $C_{2}$ is the
same inverting $z$ and $\zeta$. The estimate (3.17) implies

$\frac{d}{dt}D^{2}(z_{t},\zeta_{t})\leq\,4C\omega(D^{2}(z_{t},\zeta_{t})),\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$

Since the inequality

$y(t)\leq y(0)+4C\int_{0}^{t}\omega(y(s))\,ds$

entails that $y(t)\leq G^{-1}(G(y(0))+4Ct)$, we conclude. ∎

Proof of Theorems 1.3 and 1.4. The main theorems in the introduction are now a
straightforward consequence of the expansion control iii) in Theorem 3.1. Both
Theorems follow from the inequality (3.5) observing that
$G^{-1}(G(0)+4Ct)=G^{-1}(-\infty)=0$.  $\square$

##  4\. $\omega$-convexity of the functional

In this section we show another consequence of the EVI formulation of bounded
solutions. For the functional $\mathscr{F}_{\varepsilon,\alpha}$ the following
relaxed $\omega$-convexity along geodesics holds, see [18] for
$\omega$-convexity of functionals on measures. We assume that bounded
solutions to (1.1) (resp. (1.2) for $\varepsilon=0$) verify that for some
$T>0$

$\|\rho_{t}\|_{L^{\infty}(\mathbb{R}^{d})}\leq
R_{T}(\rho^{0},v^{0}):=R\left(T,\|\rho^{0}\|_{L^{\infty}(\mathbb{R}^{d})},\mathscr{F}_{\varepsilon,\alpha}(\rho^{0},v^{0})\right),\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T)\,.$ (4.1)

This assumption has been proved in several cases, see the introduction for
more details.

######  Theorem 4.1.

Assume that bounded solutions for the evolutions (1.1) (resp. (1.2) for
$\varepsilon=0$) exist and verify (4.1). Then, for every $z^{0},z^{1}\in
X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$ (reduced to $X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$) and every geodesic
$s\in[0,1]\to z^{s}$ of the space $(X_{\varepsilon},D)$, connecting $z^{0}$ to
$z^{1}$ there holds for all $s\in[0,1]$

$\mathscr{F}_{\varepsilon,\alpha}(z^{s})\leq(1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})+s\mathscr{F}_{\varepsilon,\alpha}(z^{1})+R_{T}\left[(1-s)\,\omega(s^{2}D^{2}(z^{0},z^{1}))+s\,\omega((1-s)^{2}D^{2}(z^{0},z^{1}))\right],$

where $R_{T}:=\max(R_{T}(z^{0}),R_{T}(z^{1}))$.

###### Proof.

We first remark that the set
$X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$ (resp. $X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$) is geodesically convex. This
is trivial for the part of the functional concerning $v$ while for the density
$\rho$ we use the classical displacement convexity of all the $L^{p}$ norms
[31]. Now, we take
$X_{\varepsilon}\cap\big{(}L^{\infty}(\mathbb{R}^{d})\times(W^{1,2}(\mathbb{R}^{d})\cap
W^{2,\infty}(\mathbb{R}^{d}))\big{)}$ (resp. $X_{0}\cap
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$) as the set of initial data for
the evolutions (1.1) (resp. (1.2)).

Consider any $\bar{z}\in
X_{\varepsilon}\cap\left(L^{\infty}(\mathbb{R}^{d})\times
W^{1,2}(\mathbb{R}^{d})\right)$ (reduced to $\bar{z}\in X_{0}\times
L^{\infty}(\mathbb{R}^{d})$ if $\varepsilon=0$). Taking into account that for
a bounded solution $t\mapsto D^{2}(z_{t},\bar{z})$ is absolutely continuous
and $t\mapsto\mathscr{F}_{\varepsilon,\alpha}(z_{t})$ is decreasing by (3.4),
from (3.3) we obtain

$\frac{1}{2}D^{2}(z_{t},\bar{z})-\frac{1}{2}D^{2}(z_{0},\bar{z})\leq\,t(\mathscr{F}_{\varepsilon,\alpha}(\bar{z})-\mathscr{F}_{\varepsilon,\alpha}(z_{t}))+C\int_{0}^{t}\omega(D^{2}(z_{r},\bar{z}))\,dr$
(4.2)

for all $t\in[0,T]$. We denote by $z_{t}^{s}$ the bounded solution of (1.1) or
(1.2) starting from the initial datum $z^{s}$. We multiply by $(1-s)$ the
inequality in (4.2) for $z_{t}=z_{t}^{s}$ and $\bar{z}=z^{0}$ and we multiply
by $s$ the inequality in (4.2) for $z_{t}=z_{t}^{s}$ and $\bar{z}=z^{1}$.
Summing up the two inequalities we obtain

$\displaystyle\frac{1}{2}((1-s)D^{2}(z^{s}_{t},z^{0})+sD^{2}(z^{s}_{t},z^{1}))$
$\displaystyle-\frac{1}{2}((1-s)D^{2}(z^{s},z^{0})+sD^{2}(z^{s},z^{1}))$
$\displaystyle\leq\,$ $\displaystyle
t((1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})+s\mathscr{F}_{\varepsilon,\alpha}(z^{1})-\mathscr{F}_{\varepsilon,\alpha}(z^{s}_{t}))$
$\displaystyle+C((1-s)\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{0}))\,dr+s\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{1}))\,dr).$

Using the fact that $s\mapsto z^{s}$ is a geodesic, the right hand side is
nonnegative, thus

$\displaystyle\mathscr{F}_{\varepsilon,\alpha}(z^{s}_{t})-(1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})$
$\displaystyle-s\mathscr{F}_{\varepsilon,\alpha}(z^{1})$ $\displaystyle\leq
C((1-s)\frac{1}{t}\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{0}))\,dr+s\frac{1}{t}\int_{0}^{t}\omega(D^{2}(z^{s}_{r},z^{1}))\,dr).$

The lower semi continuity of
$t\mapsto\mathscr{F}_{\varepsilon,\alpha}(z^{s}_{t})$ and the continuity of
$r\mapsto D^{2}(z^{s}_{r},z^{i})$, $i=0,1$ yield

$\mathscr{F}_{\varepsilon,\alpha}(z^{s})\leq(1-s)\mathscr{F}_{\varepsilon,\alpha}(z^{0})+s\mathscr{F}_{\varepsilon,\alpha}(z^{1})+C((1-s)\omega(D^{2}(z^{s},z^{0}))+s\omega(D^{2}(z^{s},z^{1}))).$

Since $s\mapsto z^{s}$ is a geodesic we have
$D^{2}(z^{s},z^{0})=s^{2}D^{2}(z^{1},z^{0})$ and
$D^{2}(z^{s},z^{1})=(1-s)^{2}D^{2}(z^{0},z^{1})$ and we conclude. ∎

##  5\. A refined result in Zygmund spaces

This section is devoted to give a rigorous justification of the estimates
stated in subsection 2.1. We will also give a slight improvement of Theorem
3.1 and Theorem 1.4 by guaranteing a suitable quasi-Lipschitz estimate under a
more general condition on the initial datum $v^{0}$. The right framework is
that of Zygmund spaces. These classes of functions were introduced in [35],
and they belong to the more general framework of Besov spaces.

#### Zygmund estimates and log-Lipschitz regularity in the elliptic case

We first introduce the basic Zygmund class $\Lambda_{1}({\mathbb{R}^{d}})$ as
the set of continuous bounded functions $f$ over $\mathbb{R}^{d}$ such that

$\sup_{x,y\in\mathbb{R}^{d}}\frac{|f(x)-2f((x+y)/2)+f(y)|}{|x-y|}<+\infty.$

It is well known that functions in the Zygmund class are in general not
Lipschitz, possibly nowhere differentiable. Indeed, functions in
$\Lambda_{1}(\mathbb{R}^{d})$ enjoy a log-Lipschitz modulus of continuity.
Therefore, for any $f\in\Lambda_{1}(\mathbb{R}^{d})$ there exists a positive
constant $C$ such that

$|f(x)-f(y)|\leq C|x-y||\log|x-y||\quad\forall x,y\in\mathbb{R}^{d},$

we refer for instance to [36, Chapter 2, §3]. More generally, following for
instance [33, Chapter 5] we may define the class $\Lambda_{n}(\mathbb{R}^{d})$
for any $n\in\mathbb{N}$ as follows. We let
$\Lambda_{0}=L^{\infty}(\mathbb{R}^{d})$ and we say that
$f\in\Lambda_{0}(\mathbb{R}^{d})$ belongs to $\Lambda_{n}(\mathbb{R}^{d})$,
$n\geq 2$, if $\nabla f\in\Lambda_{n-1}(\mathbb{R}^{d})$ or, equivalently, if
$f\in W^{n-1,\infty}(\mathbb{R}^{d})$ and all the derivatives of $f$ of order
$n-1$ belong to $\Lambda_{1}(\mathbb{R}^{d})$. In the usual notation of Besov
spaces, $\Lambda_{n}$ corresponds to $B^{n}_{\infty,\,\infty}$. In this
framework we have

Proof of Proposition 2.1. If $\alpha>0$, from the general theory on Bessel
potentials (see for instance [33, Chapter 5, §3-6]) we learn that by
convolution with the Bessel kernel $\mathcal{B}_{\alpha,d}$ we indeed get two
indices of regularity in $\Lambda_{n}$ spaces. Therefore, if $\rho\in
L^{\infty}(\mathbb{R}^{d})$, we indeed get that
$v=\mathcal{B}_{\alpha,d}\ast\rho$ belongs to $\Lambda_{2}(\mathbb{R}^{d})$,
and thus $\nabla v\in\Lambda_{1}(\mathbb{R}^{d})$ and (2.1) follows. For the
case $\alpha=0$ we address to the references mentioned in Section 2 (it is
also possible to directly check that $\nabla v\in L^{\infty}(\mathbb{R}^{d})$,
and then the Newtonian potential behaves like the Bessel potential near the
origin so that $\nabla v$ is also log-Lipschitz).  $\square$

#### Zygmund estimates and log-Lipschitz regularity in the parabolic case

Let $T>0$. In this section, let us denote $Q_{T}:=(0,T)\times\mathbb{R}^{d}$
and then $\bar{Q}_{T}:=[0,T]\times\mathbb{R}^{d}$. In the half $d+1$
dimensional space, we consider the standard parabolic metric

$\delta((x,t),(y,s)):=\max\\{|x-y|,\sqrt{|t-s|}\\}.$

With respect to the parabolic metric, the definition of Zygmund spaces adapts
as follows. We have $\Lambda_{0}(\bar{Q}_{T}):=L^{\infty}(Q_{T})$, and
$\Lambda_{1}(\bar{Q}_{T})$ is the space of continuous bounded functions $f$
over $\bar{Q}_{T}$ such that there hold

$\sup_{{x,y\in\mathbb{R}^{d}}\atop{t\in[0,T]}}\\!\\!\frac{|f(x,t)-2f((x+y)/2,t)+f(y,t)|}{|x-y|}\;+\\!\\!\\!\sup_{{x\in\mathbb{R}^{d}}\atop{0\leq
s<t\leq
T}}\\!\\!\\!\frac{|f(x,t)-2f(x,(t+s)/2)+f(x,s)|}{|t-s|^{1/2}}<+\infty.$ (5.1)

Moreover, we say that $f\in L^{\infty}(Q_{T})$ belongs to
$\Lambda_{2}(\bar{Q}_{T})$ if

$\sup_{x\in\mathbb{R}^{d}\,,\,0\leq s<t\leq
T}\frac{|f(x,t)-2f(x,(t+s)/2)+f(x,s)|}{|t-s|}<+\infty$

and $\nabla f\in\Lambda_{1}(\mathbb{R}^{d})$. In particular, we see that
$f\in\Lambda_{2}(\bar{Q}_{t})$ implies $f\in
L^{\infty}((0,T);W^{1,\infty}(\mathbb{R}^{d}))$, with $\nabla f$ satisfying
(5.1), so that finally $f$ satisfies also (2.3).

When dealing with parabolic equations, it is suitable to consider spaces of
functions defined with respect to the parabolic metric, since it is natural to
deal with functions which have derivative up to order $k$ with respect to time
and $2k$ with respect to space. For classic results, we refer for instance to
[16] or to the monograph [26], where estimates are derived in Sobolev and
Hölder spaces of this kind, see Chapter 4 therein.

In [16] we find that if the forcing term of the heat equation has bounded mean
oscillation (BMO), still with respect to the parabolic metric, than the same
holds true for second order space derivatives and first order time derivatives
of the solution. This would be enough for deducing that first derivatives in
space are in the Zygmund class with respect to the parabolic metric and that
therefore they satisfy a log-Lipschitz estimate. The results in [16] deal only
with null initial datum, but they can be generalized to more general data with
suitable regularity requirements. Some extensions involving initial data in
Zygmund classes are found in [1, 23], based on direct estimates on fundamental
solutions. Summing up, we have

Proof of Proposition 2.2. Suppose that $v$ is the solution (convolution with
fundamental operator) of the forced heat equation $\partial_{t}v=\Delta
v+\rho$. Suppose $\rho\in\Lambda_{0}(\bar{Q}_{T})$ and
$v^{0}\in\Lambda_{2}(\mathbb{R}^{d})$. Then we have
$v\in\Lambda_{2}(\bar{Q}_{T})$. See [16] for the case $v^{0}=0$, see [23,
Theorem 4] for a general result. If we consider the second equation of (1.1)
with $\alpha>0$, the fundamental solution is just multiplied by a decaying
exponential at infinity and the same result carries over. Therefore, a
sufficient condition in order to have $v$ satisfying (2.3) is $v^{0}\in
W^{2,\infty}(\mathbb{R}^{d})$, because
$W^{2,\infty}(\mathbb{R}^{d})\subset\Lambda_{2}(\mathbb{R}^{d})$.  $\square$

This gives a rigorous justification of the assumptions on the initial datum of
Theorem 1.4. However a refined analysis shows that this assumption can be
weakened, as we do next.

#### Initial datum in $\Lambda_{1}(\mathbb{R}^{d})$

We have to consider the weighted Zygmund space $\Lambda_{2}^{-1}(Q_{T})$,
defined as the corresponding space $\Lambda_{2}(\bar{Q}_{T})$, with the
addition of a time weight which is divergent as $t\to 0$. In particular,
locally in $Q_{T}$ functions in $\Lambda_{2}^{-1}(Q_{T})$ have the same
smoothness as the ones in $\Lambda_{2}(\bar{Q}_{T})$, but this regularity does
no more extend to the closure of $Q_{T}$. More precisely, by definition
$f\in\Lambda_{2}^{-1}(Q_{T})$ means that $f\in\Lambda_{1}(\bar{Q}_{T})$,

$\sup_{{x,y\in\mathbb{R}^{d}}\atop{t\in[0,T]}}\sqrt{t}\;\frac{|\nabla
f(x,t)-2\nabla f((x+y)/2,t)+\nabla f(y,t)|}{|x-y|}<+\infty$ (5.2)

and the second finite differences of $f$ and $\nabla f$ with respect to time
verify the corresponding estimates, as in the definition of
$\Lambda_{2}(\bar{Q}_{T})$, still with the addition of the weight $t^{1/2}$.

######  Theorem 5.1.

Let $T>0$. Let $\rho^{0}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and $v^{0}\in\Lambda_{1}(\mathbb{R}^{d})\cap
W^{1,2}(\mathbb{R}^{d})$. Let $z_{t}=(\rho_{t},v_{t})$ be a bounded solution
on $[0,T]\times\mathbb{R}^{d}$ to the Cauchy problem for (1.1), according to
Definition 1.2, with initial datum $z^{0}=(\rho^{0},v^{0})$. For any reference
point
$\bar{z}=(\bar{\rho},\bar{v})\in(\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d}))\times W^{1,2}(\mathbb{R}^{d})$, the general EVI
holds

$\frac{1}{2}\frac{d}{dt}D^{2}(z_{t},\bar{z})\leq\,\mathscr{F}_{\varepsilon,\alpha}(\bar{z})-\mathscr{F}_{\varepsilon,\alpha}(z_{t})+Ct^{-1/2}\omega(D^{2}(z_{t},\bar{z}))\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T),$ (5.3)

for a constant $C$ depending on
$\|\rho\|_{L^{\infty}((0,T)\times{\mathbb{R}^{d}})}$,
$\|v^{0}\|_{\Lambda^{1}(\mathbb{R}^{d})}$,
$\|\bar{\rho}\|_{L^{\infty}(\mathbb{R}^{d})}$.

Moreover the EDE (3.4) holds, and the expansion control property holds in this
form: given another bounded solution $t\mapsto\zeta_{t}$ as above with initial
datum $\zeta^{0}\in(\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d}))\times(\Lambda_{1}(\mathbb{R}^{d})\cap
W^{1,2}(\mathbb{R}^{d}))$ there is

$D^{2}(z_{t},\zeta_{t})\leq
G^{-1}(G(D^{2}(z^{0},\zeta^{0}))+8C\sqrt{t})\quad\text{for every }t\in[0,T),$
(5.4)

where $C$ is a constant depending on
$\|\rho\|_{L^{\infty}((0,T)\times\mathbb{R}^{d})}$ and
$\|v^{0}\|_{\Lambda_{1}(\mathbb{R}^{d})}$ (and the same quantities associated
to $\zeta$). In particular, $z=\zeta$ if $z^{0}=\zeta^{0}$.

###### Proof.

Since we are in the hypotheses of [23, Theorem 4], $v$ belongs to
$\Lambda_{2}^{-1}(Q_{T})$, so that (5.2) above holds for $v$ and then, due to
the log-Lipschitz regularity in the Zygmund class, we deduce

$|\nabla v_{t}(x)-\nabla v_{t}(y)|\leq Kt^{-1/2}|x-y||\log|x-y||,$ (5.5)

for all $x\in\mathbb{R}^{d}$, $t\in(0,T)$, where $K$ is a suitable constant
depending only on $T$ and the data. Notice that from the definition of
$\Lambda_{2}^{-1}(Q_{T})$, it does not follow that $\nabla v\in
L^{\infty}(Q_{T})$. However, in [23, Theorem 4] it is also shown that, still
for $v^{0}\in\Lambda_{1}(\mathbb{R}^{d})$ and $\rho\in L^{\infty}(Q_{T})$, the
solution $v_{t}$ of the parabolic equation also satisfies

$\|\nabla v_{t}(\cdot)\|_{L^{\infty}(\mathbb{R}^{d})}\leq\bar{K}(1+|\log
t|),\quad t\in(0,T),$ (5.6)

again for some positive $\bar{K}$ depending on $\|\rho\|_{L^{\infty}(Q_{T})}$
and $\|v^{0}\|_{\Lambda_{1}(\mathbb{R}^{d})}$. Taking (5.6) into account, it
is clear that (5.5) can be improved (we do not relabel the constant) into

$|\nabla v_{t}(x)-\nabla v_{t}(y)|\leq Kt^{-1/2}|x-y|(1+\log^{-}|x-y|).$

Thus we deduce the weighted analogous of (2.3), that is

$|\nabla v_{t}(x)-\nabla
v_{t}(x)|^{2}\leq\frac{C^{2}}{t}\>\varphi(|x-y|^{2}),$ (5.7)

where $C$ is a new suitable positive constant depending on the data and
$\varphi$ is defined in (2.2). Following the line of the proof Theorem 3.1 we
reach the estimate (3.8) for $II_{t}$, which now has to be changed because we
have to use (5.7), obtaining

$|II_{t}|\leq
Ct^{-1/2}W_{2}(\rho_{t},\bar{\rho})\sqrt{\mathfrak{m}\varphi(\mathfrak{m}^{-1}W_{2}^{2}(\rho_{t},\bar{\rho}))}=Ct^{-1/2}\omega(W_{2}^{2}(\rho_{t},\bar{\rho})).$

We can repeat all the other steps which lead to (3.14), obtaining the
corresponding EVI with the additional weight $t^{-1/2}$, which directly lead
to (5.3). We conclude as in Step 4 of the proof of Theorem 3.1: from (5.3),
the EDE formulation (3.4) follows, still referring to [2, Proposition 3.6].
Moreover, (5.4) follows by (5.3) by

$\frac{d}{dt}D^{2}(z_{t},\zeta_{t})\leq\,4Ct^{-1/2}\omega(D^{2}(z_{t},\zeta_{t})),\quad\text{for
$\mathcal{L}^{1}$-a.e. }t\in(0,T).$

Indeed the inequality

$y(t)\leq y(0)+4C\int_{0}^{t}s^{-1/2}\omega(y(s))\,ds$

implies that $y(t)\leq G^{-1}(G(y(0))+8C\sqrt{t})$ as desired. Finally, the
uniqueness result follows since $G(0)=-\infty$ and $G^{-1}(-\infty)=0$. ∎

##  6\. The case of nonlinear diffusion

We show next how to adapt our techniques to more general aggregation diffusion
equations in a quite straightforward way. Let us consider the problem

$\left\\{\begin{array}[]{rl}\partial_{t}\rho&=\mathrm{div}\,(\rho\nabla
P(\rho))-\,\mathrm{div}\,(\rho\nabla v),\\\\[5.69054pt]
\varepsilon\partial_{t}v&=\Delta v+\rho-\alpha v,\end{array}\right.$ (6.1)

to which we associate the functional

$\mathscr{G}_{\varepsilon,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\Psi(\rho)-v\rho)\,dx+\frac{1}{2}\int_{\mathbb{R}^{d}}(|\nabla
v|^{2}+\alpha v^{2})\,dx,$ (6.2)

for all $\varepsilon>0$, $\alpha\geq 0$,
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}({\mathbb{R}^{d}})$, $v\in W^{1,2}(\mathbb{R}^{d})$, where
${\Psi(\rho):=\int_{0}^{\rho}P(r)\,dr}$. We give the same restrictions as [3,
§9.3], the first one being

$\lim_{r\to 0}\frac{\Psi(r)}{r^{q}}>-\infty\quad\mbox{for some
}q>\frac{d}{d+2},$

a property ensuring that ${\int_{\mathbb{R}^{d}}\Psi(\rho)\neq-\infty}$.
Moreover, the crucial property to be satisfied by the new nonlinearity is the
displacement convexity, that is the map $r\mapsto r^{d}\Psi(r^{-d})$ is convex
and nondecreasing on $(0,+\infty)$. This notion, introduced in [31], is
stronger than convexity and corresponds for $C^{2}$ functions to the
inequality

$r^{-1}\Psi(r)-\Psi^{\prime}(r)+r\Psi^{\prime\prime}(r)\geq-\,\frac{1}{d-1}\,r\Psi^{\prime\prime}(r)\qquad\forall\;r\in(0,+\infty).$

The more relevant cases correspond to nonlinear diffusion of power kind.
Indeed, if

$\Psi(\rho)=\frac{1}{m-1}\,\rho^{m},\quad m\geq\frac{d-1}{d}$

the displacement convexity property holds. The case $m>1$ (resp. $m<1$)
correspond to a slow diffusion (resp. fast diffusion) in the equation. On the
other hand, the linear diffusion is recovered taking $P(\rho)=\log\rho$, it is
seen that in this case functional (6.2) is reduced, up to a constant, to
(1.3). Finally, let us mention that the free-energy functional in the
parabolic-elliptic case is similar to (1.4) and given by

$\mathscr{G}_{0,\alpha}(\rho,v):=\int_{\mathbb{R}^{d}}(\Psi(\rho)-\frac{1}{2}v\rho)\,dx,$
(6.3)

for $\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and $v=\mathcal{B}_{\alpha,d}\ast\rho$. It can be
written as (6.2), taking into account the same renormalization as in (3.1), to
be done in the pathological cases $\varepsilon=\alpha=0$ and $d=1,2$.

The notion of bounded solution is completely analogous to Definitions 1.1 and
1.2, both for the parabolic-elliptic and the parabolic-parabolic case. Indeed,
the only point to adapt is the finiteness of the Fisher information, now
rewritten into the generalized version

$\int_{0}^{T}\int_{\mathbb{R}^{d}}|\nabla
P(\rho_{t}(x))|^{2}\,\rho_{t}(x)\,dx\,dt<+\infty.$ (6.4)

######  Corollary 6.1.

Theorem 1.3, Theorem 1.4, Theorem 3.1 and Theorem 5.1 hold for bounded
solutions to (6.1).

###### Proof.

The displacement convexity property makes the internal energy functional
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\mapsto\int_{\mathbb{R}^{d}}\Psi(\rho(x))\,dx$
convex along Wasserstein geodesics, as shown in [3, §9.3]. This in turn gives
the possibility to write down a subdifferential inequality in Wasserstein
sense (for a definition see [3, §10.1.1]) as follows. Let
$\rho\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ be such that $\int_{\mathbb{R}^{d}}|\nabla
P(\rho)|^{2}\,\rho\,dx$ is finite. Then

$\int_{\mathbb{R}^{d}}\Psi(\bar{\rho}(x))\,dx-\int_{\mathbb{R}^{d}}\Psi(\rho(x))\,dx\geq\int_{\mathbb{R^{d}}}\langle\nabla
P(\rho(x)),\mathcal{T}(x)-x\rangle\,\rho(x)\,dx,$ (6.5)

for any $\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})$, where
$\mathcal{T}$ is the optimal transport map from $\rho$ to $\bar{\rho}$.
Convexity and differentiability of functionals defined on probability
densities, as the internal energy, are standard elements in the theory of
Wasserstein gradient flows. For the proof of inequality (6.5), which
characterizes the vector $\nabla P(\rho)$ as the Wasserstein subdifferential
of the internal energy functional, we refer to [2, §3.3.1] or to the general
theory in [3, §10.4.3].

On the other hand, (6.5) can be used to generalize the proof of Theorem 3.1.
Indeed, if $(\rho_{t},v_{t})$ solves (6.1) according to our notion of
solution, thanks to (6.4) $\rho_{t}$ satisfies the identity (6.5) for almost
any $t$. From this inequality, all the rest of the proof of Theorem 3.1 can be
carried out. Indeed, with the same notation therein, we obtain the
$\mathcal{L}^{1}$-a.e. $t\in(0,T)$ inequality

$\displaystyle I_{t}:=\mathscr{G}_{\varepsilon,\alpha}(\bar{\rho},\bar{v})$
$\displaystyle-\mathscr{G}_{\varepsilon,\alpha}(\rho_{t},v_{t})+\int_{\mathbb{R}^{d}}(\bar{v}(x)-v_{t}(x))\bar{\rho}(x)\,dx\geq$
$\displaystyle\int_{\mathbb{R}^{d}}\langle\rho_{t}(x)\nabla
P(\rho_{t}(x))-\rho_{t}(x)\nabla
v(x),\mathcal{T}_{t}(x)-x\rangle\,dx-C\omega(W_{2}^{2}(\rho_{t},\bar{\rho})),$

for any $\bar{\rho}\in\mathscr{M}_{2}(\mathbb{R}^{d};\mathfrak{m})\cap
L^{\infty}(\mathbb{R}^{d})$ and any $\bar{v}\in W^{1,2}(\mathbb{R}^{d})$ if
$\varepsilon>0$ or $\bar{v}=\mathcal{B}_{\alpha,d}\ast\bar{\rho}$ if
$\varepsilon=0$. This estimate substitutes (3.9) in the proof of Theorem 3.1.
The rest of the proofs is completely analogous. ∎

### Acknowledgements

The authors would like to thank Paolo Acquistapace and Lucilla Corrias for
several discussions about this work. JAC acknowledges support from the project
MTM2011-27739-C04-02 DGI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de
Catalunya. JAC acknowledges support from the Royal Society through a Wolfson
Research Merit Award. This work was partially supported by Engineering and
Physical Sciences Research Council grant number EP/K008404/1. SL and EM has
been partially supported by the INDAM-GNAMPA project 2011 ”Measure solution of
differential equations of drift-diffusion, interactions and of Cahn-Hilliard
type”. EM has been partially supported by a postdoctoral scholarship of the
Fondation Mathématique Jacques Hadamard, he acknowledges hospitality from
Paris-Sud University.

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