On Patlak-Keller-Segel system for several populations: a gradient flow approach

TL;DR

This paper investigates the global existence of solutions for a multi-species Patlak-Keller-Segel system using a gradient flow framework in Wasserstein space, establishing conditions for solutions and their energy dissipation.

Contribution

It introduces a gradient flow approach in Wasserstein space to prove global existence of solutions for multi-species Patlak-Keller-Segel systems under sub-critical initial mass conditions.

Findings

Solutions exist globally under sub-critical mass conditions

The system satisfies an energy dissipation inequality

Gradient flow structure aids in analyzing solution existence

Abstract

We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.

Equations

$$\{\begin{array}{ll}& {\partial }_t{\rho }_i(x,t)={\Delta }_x{\rho }_i(x,t)-{\sum }_{j=1}^n{a}_{ij}{\nabla }_x\cdot \left({\rho }_i(x,t){\nabla }_x{u}_j(x,t)\right),in{\mathbb{R}}^2\times (0,\infty ),\\ & -{\Delta }_x{u}_i(x,t)={\rho }_i(x,t),in{\mathbb{R}}^2\times (0,\infty ),\\ & {\rho }_i(x,0)={\rho }_i^0,i=1,\dots ,n,\end{array}$$

$$u_i(x,t)=-\frac{1}{2\pi }{\int }_{{\mathbb{R}}^2}\ln |x-y|{\rho }_i(y,t)dy,i=1,\dots ,n.$$

$${\int }_{{\mathbb{R}}^2}\mathbf{\rho }(x,t)dx={\int }_{{\mathbb{R}}^2}{\mathbf{\rho }}^0(x)dx=\mathbf{\beta },forallt>0.$$

$$\mathcal{F}(\mathbf{\rho }(\cdot ,t))+{\int }_0^t{\mathcal{D}}_{\mathcal{F}}(\mathbf{\rho }(\cdot ,s))ds=\mathcal{F}({\mathbf{\rho }}^0),$$

$$\mathcal{F}(\mathbf{\rho })=\sum _{i=1}^n{\int }_{{\mathbb{R}}^2}{\rho }_i(x)\ln {\rho }_i(x)dx+\sum _{i=1}^n\sum _{j=1}^n\frac{a_{ij}}{4\pi }{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\rho }_i(x)\ln |x-y|{\rho }_j(y)dxdy,$$

$${\mathcal{D}}_{\mathcal{F}}(\mathbf{\rho })=\sum _{i=1}^n{\int }_{{\mathbb{R}}^2}{\left|\frac{\nabla {\rho }_i(x)}{{\rho }_i(x)}-\sum _{j=1}^n{a}_{ij}\nabla u_j(x)\right|}^2{\rho }_i(x)dx.$$

$$M_2(\mathbf{\rho }(\cdot ,t))dx=\frac{{\Lambda }_I(\mathbf{\beta })}{2\pi }t+M_2({\mathbf{\rho }}^0),$$

$${\Lambda }_J(\mathbf{\beta }):=\sum _{i\in J}{\beta }_i\left(8\pi -\sum _{j\in J}{a}_{ij}{\beta }_j\right),forall\varnothing \ne J\subset I.$$

$${\mathcal{F}}_{scalar}(\rho ):={\int }_{{\mathbb{R}}^2}\rho (x)\ln \rho (x)dx+\frac{a}{4\pi }{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\rho (x)\ln |x-y|\rho (y)dxdy\ge -C$$

$${\partial }_t{\rho }_i=\nabla \cdot \left({\rho }_i\nabla \frac{\delta \mathcal{F}}{\delta {\rho }_i}(\mathbf{\rho })\right),i=1,\dots ,n.$$

$${\Gamma }^{\mathbf{\beta }}=\{\mathbf{\rho }=({\rho }_i{)}_{i=1}^n|{\rho }_i\in L_{+}^1({\mathbb{R}}^2),{\int }_{{\mathbb{R}}^2}{\rho }_i(x)\ln {\rho }_i(x)dx<+\infty ,{\int }_{{\mathbb{R}}^2}{\rho }_i(x)dx={\beta }_i,$$

$${\int }_{{\mathbb{R}}^2}{\rho }_i(x)\ln (1+|x{|}^2)dx<+\infty \}$$

$${\mathbf{\rho }}_{\tau }^k\in \arg \underset{\mathbf{\rho }\in {\Gamma }_2^{\mathbf{\beta }}}{\min }\left(\mathcal{F}(\mathbf{\rho })+\frac{1}{2\tau }{\mathbf{d}}_{\text{mbox}W}^2(\mathbf{\rho },{\mathbf{\rho }}_{\tau }^{k-1})\right),k\ge 1$$

$${\Gamma }_2^{\mathbf{\beta }}:=\left\{\mathbf{\rho }\in {\Gamma }^{\mathbf{\beta }}|M_2(\mathbf{\rho }):=\sum _{i=1}^n{\int }_{{\mathbb{R}}^2}|x{|}^2{\rho }_{i}dx<+\infty \right\},$$

$$\{\begin{array}{l}{\Lambda }_J(\mathbf{\beta })\ge 0,forall\varnothing \ne J\subset \{1,\dots ,n\},\\ \text{mbox}ifforsomeJ,{\Lambda }_J(\mathbf{\beta })=0,then{a}_{ii}+{\Lambda }_{J\backslash \{i\}}(\mathbf{\beta })>0,\forall i\in J\end{array}$$

$${\Lambda }_J(\mathbf{\beta })>0,forall\varnothing \ne J\subset I.$$

$$\displaystyle\mathcal{A}_{T}(\bm{\rho}):=\sup_{t\in[0,T]}\Big{(}\mathcal{H}_{+}(\bm{\rho}(t))+M_{2}(\bm{\rho}(t))\Big{)}+\int_{0}^{T}\mathcal{D}_{\mathcal{F}}(\bm{\rho}(t))\ dt<+\infty,\ \forall\ T\in(0,T^{*})$$

$${\int }_0^{T^{\ast }}{\int }_{{\mathbb{R}}^2}{\partial }_t\xi (x,t){\rho }_i(x,t)dxdt+{\int }_{{\mathbb{R}}^2}\xi (x,0){\rho }_i^0(x)dx$$

$$-{\int }_0^{T^{\ast }}{\int }_{{\mathbb{R}}^2}{\rho }_i(x,t)\left(\frac{{\nabla }_x{\rho }_i(x,t)}{{\rho }_i(x,t)}-\sum _{j=1}^n{a}_{ij}{\nabla }_x{u}_j(x,t)\right)\cdot {\nabla }_x\xi (x,t)dxdt=0$$

$$\frac{\mathcal{H}({\mathbf{\rho }}_{ϵ})-\mathcal{H}({\mathbf{\rho }}_{\tau }^k)}{ϵ},\frac{{\mathbf{d}}_{\text{mbox}W}^2({\mathbf{\rho }}_{ϵ},{\mathbf{\rho }}_{\tau }^{k-1})-{\mathbf{d}}_{\text{mbox}W}^2({\mathbf{\rho }}_{\tau }^k,{\mathbf{\rho }}_{\tau }^{k-1})}{2ϵ}$$

$$\frac{1}{ϵ}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\left(\ln |x-y+ϵ({\zeta }_i(x)-{\zeta }_j(y))|-\ln |x-y|\right){\rho }_{\tau ,i}^k(x){\rho }_{\tau ,j}^k(y)dxdy.$$

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{({\zeta }_i(x)-{\zeta }_i(y))\cdot (x-y)}{|x-y{|}^2}{\rho }_{\tau ,i}^k(x){\rho }_{\tau ,i}^k(y)dxdy,$$

$$\sum _{i=1}^n{\int }_0^T{\int }_{{\mathbb{R}}^2}{\left|\frac{\nabla {\rho }_i(x,t)}{{\rho }_i(x,t)}\right|}^2{\rho }_i(x,t)dxdt<+\infty .$$

$$\sum _{i=1}^n{\int }_0^T{\int }_{{\mathbb{R}}^2}{\left|\frac{\nabla {\rho }_i(x,t)}{{\rho }_i(x,t)}-\sum _{j=1}^n{a}_{ij}\nabla u_j(x,t)\right|}^2{\rho }_i(x,t)dxdt+\mathcal{F}(\mathbf{\rho }(T))\le \mathcal{F}({\mathbf{\rho }}^0).$$

$${\int }_{{\mathbb{R}}^2}\psi (x)d\nu (x)={\int }_{{\mathbb{R}}^2}\psi (T(x))d\mu (x),forevery\psi \in L^1({\mathbb{R}}^2,d\nu ).$$

$$d_{\text{mbox}W}^2(\mu ,\nu ):=\underset{\pi \in \Pi (\mu ,\nu )}{\inf }\left[{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|x-y{|}^2\pi (dxdy)\right],$$

$$\Pi (\mu ,\nu ):=\{\pi \in \mathcal{P}({\mathbb{R}}^2\times {\mathbb{R}}^2)|(P_1)\#\pi =\mu ,(P_2)\#\pi =\nu \},$$

$$d_{\text{mbox}W}^2(\mu ,\nu )={\int }_{{\mathbb{R}}^2}|x-\nabla \phi (x){|}^{2}d\mu (x),where\nabla \phi \#\mu =\nu .$$

$$d_{\text{mbox}W}(\mu ,\nu )={\beta }^{\frac{1}{2}}{d}_{\text{mbox}W}\left(\frac{\mu }{\beta },\frac{\nu }{\beta }\right).$$

$$d_{\text{mbox}W}^2(\mu ,\nu )={\int }_{{\mathbb{R}}^2}|x-\nabla \phi (x){|}^{2}d\mu (x),$$

$${\int }_{{\mathbb{R}}^2}U(g(x))dx={\int }_{{\mathbb{R}}^2}U\left(\frac{f(x)}{de{t}_A{D}^2\phi (x)}\right)de{t}_A{D}^2\phi (x)dx.$$

Full text

1. 1 Introduction
2. 2 Notations and Preliminaries
3. 3 Minimizing Movement Scheme
4. 4 A priori Estimates and the regularity of minimizers
5. 5 The Euler Lagrange equation
6. 6 Estimates on the time interpolation
7. 7 Convergence of the MM scheme
8. 8 Free energy inequality
9. 9 Appendix

1. Introduction

In this article, we study the global existence and uniqueness of solutions to the parabolic-elliptic Patlak-Keller-Segel system (PKS-system in short) for $n$ populations interacting via a self produced chemical agent on the two-dimensional Euclidean space $\mathbb{R}^{2}.$ The evolution of these living cells are governed by the following system of equations

[TABLE]

where $\rho_{i}(x,t)$ denotes the cell density of the $i$-th population, $u_{i}(x,t)$ denotes the concentration of the chemical, called chemoattractant, produced by the $i$-th population and $a_{ij}$ are constants denoting the sensitivity of the $i$-th population towards the chemical gradient produced by the $j$-th population and $\rho_{i}^{0}$ is the initial cell distribution of the $i$-th population. Since the solutions to the Poisson equation $-\Delta u=\rho$ is unique up to a harmonic function, we define concentration of the chemoattractant $u_{i}$ by the Newtonian potential of $\rho_{i}$

[TABLE]

The sensitivity parameters $a_{ij}>0$ meaning that the $i$-th population is attracted to the chemoattractant produced by the $j$-th population and tends to climb up its gradient. On the other hand, $a_{ij}<0$ meaning that $i$-th population is repelled from the same and tends to climb down its gradient. In particular, $a_{ii}>0$ (or $a_{ii}<0$) is the condition of self-attraction (or, self-repulsion) of the population $i.$ The case $a_{ij}a_{ji}<0$ is the unhappy situation between the $i$-th and $j$-th population and is the origin of conflict of interests. In this article, we assume the sensitivity matrix $(a_{ij})$ is symmetric with non-negative entries $a_{ij}\geq 0,\ for\ all\ i,j$, that is the conflict free case.

The manifestation of a single population or the scalar case $n=1$ (where $a:=a_{11}$) has been the subject of intensive research over the past couple of decades. See [Pat53, KS70, Wol02, Hor03, Hor11] for the biological motivations.

Equation (1.1) is a typical example of a conservative drift-diffusion equation. The smoothing effect induced by the diffusion term $\Delta\rho_{i}$ and the weighted cumulative drift induced by the chemical gradients $\sum_{j=1}^{n}a_{ij}\nabla u_{j}$ which assists the cells to accumulate, compete against each other. It is well understood, at least for the scalar case, the $L^{1}$-norm of the initial datum is a salient parameter which separates the dichotomy between the global in time existence and the chemotactic collapse (or, finite time blow up). More precisely, if the initial number of bacteria is smaller than the critical threshold $\int_{\mathbb{R}^{2}}\rho_{0}<8\pi/a$ then the process of aggression counterbalanced by the diffusion [BDP06]. However, if it crosses the critical threshold, i.e., $\int_{\mathbb{R}^{2}}\rho^{0}>8\pi/a$ the production of the chemical agent attracting the cells increase so as that the diffusion can no longer compete against the drift force, resulting in an inevitable chemotactic collapse. The critical case $\int_{\mathbb{R}^{2}}\rho^{0}=8\pi/a$ is the perfect balance between these two opposing forces. In this case, a solution exists globally in time [BCM08, BKLN06b], but if the second moment of the initial data is finite, then the solutions concentrate in the form of a Dirac delta measure as time $t$ goes to infinity [BCM08]. We refer the interested readers to [CP81, NS98, SS02, SS04, Suz05, BKLN06a, BKLN06b, BDP06, BCM08, BDEF10, BCC12, CD14, FM16] and the references therein devoted to the study of parabolic-elliptic PKS-system for single population, and also excellent survey articles [Hor03, Hor04, Bla13] in this regard.

A solution $\bm{\rho}:=(\rho_{1},\ldots,\rho_{n})$ to the PKS-system (1.1), at least formally, possess the following fundamental identities:

• •

Conservation of mass:

[TABLE]

• •

Free energy dissipation or the free energy identity:

[TABLE]

where the free energy $\mathcal{F}$ is defined by

[TABLE]

and the dissipation of free energy $\mathcal{D}_{\mathcal{F}}$ is defined by

[TABLE]

Moreover, if the second moment of the initial condition $M_{2}(\bm{\rho}^{0}):=\sum_{i=1}^{n}\int_{\mathbb{R}^{2}}|x|^{2}\rho^{0}_{i}(x)\ dx$ is finite then formally

[TABLE]

where $I=\{1,\ldots,n\}$ and $\Lambda_{I}(\bm{\beta})$ is a quadratic polynomial in $\bm{\beta}$ defined by

[TABLE]

In particular, when $n=1,\ \Lambda_{\{1\}}(\beta)=\beta(8\pi-a_{11}\beta).$ The critical constant $8\pi$ in single population emerges from the time evolution of the second moment (1.7). However, the proof of existence of global in time solution in the sub-critical case is much more delicate issue and has been explored in [BDP06] using the energy method. One of the fundamental tool in their analysis is the logarithmic Hardy-Littlewood-Sobolev (HLS) inequality in $\mathbb{R}^{2}:$

[TABLE]

for all $\rho\in\Upsilon^{\beta}:=\{\rho\in L^{1}_{+}(\mathbb{R}^{2})|\ \int_{\mathbb{R}^{2}}\rho\ln\rho<+\infty,\int_{\mathbb{R}^{2}}\ln(1+|x|^{2})\rho<+\infty,\int_{\mathbb{R}^{2}}\rho=\beta\}$ if and only if $\beta=8\pi/a.$ Roughly speaking, if $\int_{\mathbb{R}^{2}}\rho_{0}<8\pi/a$ and the total mass ($\int_{\mathbb{R}^{2}}\rho(x,t)\ dx$) being conserved for all time $t,$ the logarithmic HLS-inequality gives an bound on the entropy of the solution ($\int_{\mathbb{R}^{2}}\rho\ln\rho$), which prevents the solutions to blow-up in finite time.

Of particular, our interest lies to the alternative observation by Otto [Ott98] who saw that these class of PDEs (1.1) inherits gradient flow structure in the space of probability measures with respect to an appropriate metric. In their subsequent works, Jordan-Kinderlehrer and Otto implemented this idea in the context of the heat and the Fokker-Planck equation [JKO98].

We observe formally that the system (1.1) can be written as

[TABLE]

This is the formal structure of a gradient flow of the free energy $\mathcal{F}$ in the space $\mathcal{P}^{\beta_{1}}(\mathbb{R}^{2})\times\cdots\times\mathcal{P}^{\beta_{n}}(\mathbb{R}^{2})$ equipped with the 2-Wasserstein distance $\bm{d}_{\mbox{{\tiny{W}}}}$ (see section $2$ for definition), where $\mathcal{P}^{\beta_{i}}(\mathbb{R}^{2})$ denotes the space of non-negative Borel measures on $\mathbb{R}^{2}$ with total mass $\beta_{i}$ and $\frac{\delta\mathcal{F}}{\delta\rho_{i}}$ denotes the first variation of the functional $\mathcal{F}$ with respect to the variable $\rho_{i}.$ The functional $\mathcal{F}$ on the product space $\mathcal{P}^{\beta_{1}}(\mathbb{R}^{2})\times\cdots\times\mathcal{P}^{\beta_{n}}(\mathbb{R}^{2})$ is defined by $\mathcal{F}(\bm{\rho})$ if $\bm{\rho}\in\Gamma^{\bm{\beta}},$ where

[TABLE]

and $+\infty$ elsewhere.

The study of gradient flows in general metric spaces is substantially a vast subject and have been pioneered by Ambrosio, Gigli and Savaré in their book [AGS05]. However, to make sense of gradient flows in a general metric space and establishing a complete existence-uniqueness theory requires a certain convexity assumption on the functional (for example $\lambda$-geodesic convexity) as well as on the metric (for example $C^{2}G^{2}$-condition). The functional $\mathcal{F}$ does not possess such convexity property (it is neither convex in the usual sense nor, displacement convex in the sense of McCann [McC97]) and hence different approach is necessary. Instead, we will rely on the time-discretized variational formulation introduce in [JKO98], known as minimizing movement scheme or, JKO-scheme and the functional analytic framework to study the convergence of the scheme: for a time step $\tau>0,$ define

[TABLE]

with $\bm{\rho}^{0}_{\tau}=\bm{\rho}^{0},$ where

[TABLE]

and throughout this article we will assume the initial condition $\bm{\rho}^{0}\in\Gamma^{\bm{\beta}}_{2}.$

The existence of a minimizer to (1.9) depends on several relations depending on $\bm{\beta}$ and the interaction matrix $(a_{ij}).$ More precisely, the boundedness from below of $\mathcal{F}$ and the functional $\bm{\rho}\longmapsto\mathcal{F}(\bm{\rho})+\frac{1}{2\tau}\bm{d}_{\mbox{{\tiny{W}}}}^{2}(\bm{\rho},\bm{\eta})$ depends on the the following optimal relations:

[TABLE]

where $\Lambda_{J}({\bm{\beta}})$ is defined by (1.8). In particular, it is shown in [CSW97, SW05] that $\Lambda_{I}(\bm{\beta})=0$ and (1.10) is necessary and sufficient condition for the boundedness from below of $\mathcal{F}$ over $\Gamma^{\bm{\beta}}.$ On the other hand, if $\bm{\eta}\in\Gamma^{\bm{\beta}}_{2},$ then (1.10) is necessary and sufficient for $\mathcal{F}(\cdot)+\frac{1}{2\tau}\bm{d}_{\mbox{{\tiny{W}}}}^{2}(\cdot,\bm{\eta})$ to be bounded from below on $\Gamma^{\bm{\beta}}_{2}.$

The existence of minimizers in (1.9) is, however, a delicate question. To give a glimpse of this question let us mention some results from [KW]. In [KW] we considered the case $\bm{\eta}=(\beta_{1}\delta_{v_{1}},\ldots,\beta_{n}\delta_{v_{n}})$ where $v_{i}$ are arbitrary vectors in $\mathbb{R}^{2}$ and $\delta_{v_{i}}$ is the Dirac measure at $v_{i}.$ We found that if $\bm{\beta}$ is sub-critical, that is $\Lambda_{J}(\bm{\bm{\beta}})>0$ for all $J$ then there always exists a minimizer. However, if $\bm{\beta}$ is critical, that is $\Lambda_{I}(\bm{\beta})=0$ and $\Lambda_{J}(\bm{\beta})>0$ for all $J\neq I,$ then the existence of a minimizer depends on the gap between $v_{i}$s. In particular, for $n=2$ and $\bm{\beta}$ critical we showed that if $v_{1}=v_{2}$ then no minimizer exists, on the other hand if $|v_{1}-v_{2}|$ is large enough then a minimizer does exist. The optimal gap for which a minimizer exists is still an open question. However, we do not address such questions in this article and only consider the case $\bm{\beta}$ sub-critical.

Definition 1.1** (Sub-critical).**

$\bm{\beta}$ is said to be sub-critical if and only if

[TABLE]

Before proceeding further let us first introduce the appropriate notion of a weak solution to the PKS-system (1.1). Throughout this article, we use the notation $\mathcal{H}(\bm{\rho}):=\sum_{i=1}^{n}\int_{\mathbb{R}^{2}}\rho_{i}\ln\rho_{i}$ to denote the entropy of the solutions and $\mathcal{H}_{+}(\bm{\rho}):=\sum_{i=1}^{n}\int_{\mathbb{R}^{2}}\rho_{i}(\ln\rho_{i})_{+}$ is the positive part of the entropy.

Definition 1.2**.**

For any initial data $\bm{\rho}^{0}$ in $\Gamma^{\bm{\beta}}_{2}$ and $T^{*}>0$ we say that a nonnegative vector valued function $\bm{\rho}\in\left(C([0,T^{*});\mathcal{D}^{\prime}(\mathbb{R}^{2}))\right)^{n}$ satisfying

[TABLE]

is a weak solution to the PKS-system (1.1) on the time interval $(0,T^{*})$ associated to the initial condition $\bm{\rho}^{0}$ if $\bm{\rho}$ satisfies (1.3) and

[TABLE]

for all $\xi\in C_{c}^{2}([0,T^{*})\times\mathbb{R}^{2})$ and for all $i=1,\ldots,n.$ If $T^{*}=+\infty$ we say $\bm{\rho}$ is a global weak solution of the system.

Notably, by virtue of finite energy dissipation and Cauchy-Schwartz inequality all the terms in the weak formulation of (1.1) makes sense. The gradient flow structure for a single population PKS-system $n=1$ with sub-critical mass has been explored earlier in [BCC08]. However, there is an essential difference between the gradient flow structure of the single population and that of multi-population, as explained below:

The first difficulty arises in pursuance of the Euler-Lagrange equation for the variational scheme (1.9). For the minimization problem of type (1.9), is standard (in the theory of optimal transport) to consider the variation $\bm{\rho}_{\epsilon}=\bm{T}_{\epsilon}\#\bm{\rho}_{\tau}^{k}$ of the minimizer $\bm{\rho}_{\tau}^{k}$ and compute $\lim_{\epsilon\rightarrow 0+}\frac{1}{\epsilon}\left(\mathcal{F}(\bm{\rho}_{\epsilon})-\mathcal{F}(\bm{\rho}_{\tau}^{k})+\frac{1}{2\tau}(\bm{d}_{\mbox{{\tiny{W}}}}^{2}(\bm{\rho}_{\epsilon},\bm{\rho}_{\tau}^{k-1})-\bm{d}_{\mbox{{\tiny{W}}}}^{2}(\bm{\rho}_{\tau}^{k},\bm{\rho}_{\tau}^{k-1}))\right)\geq 0,$ where $\bm{T}_{\epsilon}$ is a $C^{\infty}$ diffeomorphism of $(\mathbb{R}^{2})^{n}$ and $\bm{T}_{\epsilon}\#$ denotes the push forward of a measure under $\bm{T}_{\epsilon}.$ In particular, we are interest in $\bm{T}_{\epsilon}$ of the form $\bm{T}_{\epsilon}=\bm{I}+\epsilon\bm{\zeta},\ \bm{\zeta}=(\zeta_{1},\ldots,\zeta_{n})$ and $\zeta_{i}\in C_{c}^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2}).$ This is where the first major difficulty arises. The minimizing movement scheme contains terms of three types: the entropy term, interaction energy term and the Wasserstein distance term. For $\bm{T}_{\epsilon}$ is being chosen of this particular form, we can evaluate the limits

[TABLE]

where $\epsilon\rightarrow 0+$. However, the interaction energy contains terms of the form

[TABLE]

To see the difficulty, for $i\neq j$ we choose $\zeta_{j}\equiv 0$ to obtain formally in the limit (up to a constant factor times) $\int_{\mathbb{R}^{2}}\zeta_{i}(x)\cdot\nabla u_{\tau,j}^{k}(x)\rho_{\tau,i}^{k}(x),$ as $\epsilon\rightarrow 0+$. The a-priori bound on the entropy of $\rho_{\tau,i}^{k}$ implies that the Newtonian potentials $u_{\tau,j}^{k}$ belongs to $H^{1}_{loc}(\mathbb{R}^{2})$ only, and so this limit is not properly defined. On the other hand if $i=j$ (as in the case $n=1$) then it is easy to pass to the limit $\epsilon\rightarrow 0+$

[TABLE]

which is well defined because $\zeta_{i}$ is smooth and compactly supported. To make sense of the terms corresponding to $i\neq j,$ we need higher regularity of the solution $\rho_{\tau,i}^{k}.$ In precise, $\rho_{\tau,i}^{k}$ must be at least $L^{2}(\mathbb{R}^{2})$ regular. We adopt the flow interchange technique introduced by Matthes, McCann and Savaré [MMS09] in spirit of [BL13, BCK*+*15] to tackle this issue. The next difficulty arises in passing to a limit in the Euler-Lagrange equation mentioned above. For the convergence of the scheme, we need a uniform in time estimates on the Newtonian potentials. It follows from the scheme that the time interpolates have $\frac{1}{2}$-Hölder estimates in time with respect to the Wasserstein distance. The Hölder regularity only gives the continuity estimates on the harmonic part of its Newtonian potential, but not on the Newtonian potential. We resolve this issue by using a refined Arzelà-Ascoli’s lemma combined with the uniform decay estimates which guarantee the convergence of interaction energy terms for all time.

Before concluding the introduction, we would like to refer the interested readers to [BL13, BCK*+*15] for related articles in the context of parabolic-parabolic PKS system of singles populations and to [EASV09, EASV10, CEV11, BG12, EVC13] for the two component chemotactic systems. For a profound discussions on the gradient flow in Wasserstein space and its application to a large class of PDEs, we refer to [Vil03, AGS05, San15].

1.1. Main results

The main result of this article is the following:

Theorem 1.3**.**

Assume $\bm{\rho}^{0}\in\Gamma^{\bm{\beta}}_{2}$ and $\bm{\beta}$ is sub-critical. Then the PKS-system (1.1) admits a global weak solution $\bm{\rho}$ in the sense of definition 1.2 with initial data $\bm{\rho}^{0}.$ Moreover, $\bm{\rho}$ satisfies for every $T>0$

• (a)

$\bm{\rho}\in\left(L^{2}((0,T)\times\mathbb{R}^{2})\right)^{n}\cap\left(L^{1}(0,T;W^{1,1}(\mathbb{R}^{2}))\right)^{n},$* and Fischer information bound:*

[TABLE]

• (b)

Free energy inequality:

[TABLE]

• (c)

The weak solution obtained in (a) is unique.

We divide the article into the following sections: In section $2$ we introduce a few notations used throughout this article and recall a few known properties of the Wasserstein distance and the free energy functional $\mathcal{F}.$ In section $3$ we propose the time-discretized JKO-scheme (see (3.1)) and prove its well-definedness. Section 4 is devoted to the a priori estimates and regularity estimates for the discrete time interpolates, which is a crucial step towards establishing the Euler Lagrange equation obtained in Section 5. In section 6 we define the time interpolation and prove the a priori estimates and regularity results as a byproduct of the results obtained in section 4. In section 7 we prove the convergence of the scheme and hence obtain the existence of a solution to the PKS-system (1.1). Finally, in section $8,$ we show that the obtained solution satisfies the free energy inequality Theorem 1.3(b) and prove the uniqueness result.

Acknowledgement: D. Karmakar is partially supported by Technion fellowship grant.

2. Notations and Preliminaries

2.1. Wasserstein distance

In this section, we recall the definition of Wasserstein 2-distance (also called the Monge-Kantorovich distance of order $2$) and some of its well-known properties.

Let $\mathcal{P}(\mathbb{R}^{2})$ be the space of all Borel probability measures on $\mathbb{R}^{2},$ $\mathcal{P}_{2}(\mathbb{R}^{2})$ denotes the subset of $\mathcal{P}(\mathbb{R}^{2})$ having finite second moments and $\mathcal{P}_{ac,2}(\mathbb{R}^{2})$ denotes the subset of $\mathcal{P}_{2}(\mathbb{R}^{2})$ which are absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{2}.$

Given two elements $\mu,\nu$ of $\mathcal{P}(\mathbb{R}^{2})$ and a map $T:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},$ we say $T$ pushes forward $\mu$ to $\nu,$ denoted by $T\#\mu=\nu,$ if for every Borel measurable subset $U$ of $\mathbb{R}^{2},$ $\nu(U)=\mu(T^{-1}(U)).$ Equivalently,

[TABLE]

On $\mathcal{P}_{2}(\mathbb{R}^{2})$ we can define a distance $d_{\mbox{{\tiny{W}}}},$ using the Monge-Kantorovich transportation problem with quadratic cost function $c(x,y)=|x-y|^{2}.$ More precisely, given $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{2})$ define

[TABLE]

where

[TABLE]

is the set of transport plans and $P_{i}:\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ denotes the canonical projections on the $i$-th factor.

The celebrated theorem of Brenier [Bre91] asserts that: if $\mu\in\mathcal{P}_{ac,2}(\mathbb{R}^{2})$ then there exists a unique (up to additive constants) convex, lower semi continuous function $\varphi$ such that $\nabla\varphi\#\mu=\nu$ and the optimal transference plan $\hat{\pi}$ on the right hand side of (2.2) is given by $\hat{\pi}=(Id,\nabla\varphi)\#\mu,$ where $Id:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is the identity mapping (see [Vil03, Theorem $2.12$]). As a consequence, we have

[TABLE]

Note that if $\mu,\nu$ are two non-negative measures on $\mathbb{R}^{2}$ (not necessarily probability measures) satisfying the total mass compatibility condition $\mu(\mathbb{R}^{2})=\nu(\mathbb{R}^{2})(=\beta>0),$ then we can also define the Wasserstein $2$-distance between them as follows:

[TABLE]

We will denote by $\mathcal{P}^{\beta}(\mathbb{R}^{2})$ the space of non-negative Borel measures with total mass $\beta$ and $\mathcal{P}^{\beta}_{2}(\mathbb{R}^{2})$ and $\mathcal{P}^{\beta}_{ac,2}(\mathbb{R}^{2})$ are defined analogously. We will also use the bold $\bm{\beta}$ notation in $\mathcal{P}^{\bm{\beta}}(\mathbb{R}^{2})$ to denote the product space $\mathcal{P}^{\beta_{1}}(\mathbb{R}^{2})\times\cdots\times\mathcal{P}^{\beta_{n}}(\mathbb{R}^{2}).$

One advantage of defining the Wasserstein distance on $\mathcal{P}^{\beta}(\mathbb{R}^{2})$ by (2.4) is that, if $\mu$ is absolutely continuous with respect to the Lebesgue measure and if $\nabla\varphi$ is the gradient of a convex function pushing $\mu/\beta$ forward to $\nu/\beta$ then $\nabla\varphi\#\mu=\nu$ and

[TABLE]

where note that $d_{\mbox{{\tiny{W}}}}(\mu,\nu)$ is defined by (2.4).

2.2. Change of variable formula

Let $\mu,\nu\in\mathcal{P}_{ac,2}(\mathbb{R}^{2})$ be two probability measures and let $f$ and $g$ be their respective densities. Let $\varphi$ be a convex function such that $\nabla\varphi\#fdx=gdx.$ $\varphi$ being convex, by Aleksandrov’s theorem, it is twice differentiable almost everywhere in its domain of definition. Let $D^{2}\varphi$ denotes the Hessian matrix of $\varphi$ (which is well defined a.e. on $\{\varphi<+\infty\}$) and let $det_{A}D^{2}\varphi$ be the determinant of $D^{2}\varphi.$ Here we use the notation $det_{A}$ because of fact that $det_{A}D^{2}\varphi$ is the absolutely continuous part of the Hessian measure $det_{H}D^{2}\varphi.$ The following change of variable formula can be found in [McC97] (see also Villani [Vil03, Theorem $4.8$]):

For every measurable function $U:[0,\infty)\rightarrow\mathbb{R}$ such that $U$ is bounded from below and $U(0)=0$

[TABLE]

In particular, choosing $U(t)=t\ln t$ and $f,g$ any non-negative densities satisfying mass compatibility condition $\int_{\mathbb{R}^{2}}f(x)dx=\int_{\mathbb{R}^{2}}g(x)dx=\beta$ with $\nabla\varphi\#fdx=gdx$ then

[TABLE]

2.3. Properties of the free energy functional

We end this section by recalling a few well known properties of the free energy functional $\mathcal{F}$ whose proof can be found in [SW05, KW].

Proposition 2.1**.**

The followings hold:

• (a)

$\mathcal{F}$* is bounded from below on $\Gamma^{\bm{\beta}}$ if and only if $\bm{\beta}$ satisfies*

[TABLE]

• (b)

For any $n$-numbers $\alpha_{i}>0,$ the functional

[TABLE]

is bounded from below on $\Gamma^{\bm{\beta}}_{2}$ if and only if $\bm{\beta}$ satisfies (1.10).

• (c)

The functionals $\mathcal{F}$ and $\mathcal{F}_{\bm{\alpha}}$ are sequentially lower semi-continuous with respect to the weak topology of $L^{1}(\mathbb{R}^{2}).$

• (d)

If $\bm{\beta}$ if sub-critical and $\min_{i\in I}\alpha_{i}>0,$ then all the sub-level sets $\{\mathcal{F}_{\bm{\alpha}}\leq C\}$ are sequentially precompact with respect to the weak topology of $L^{1}(\mathbb{R}^{2}).$

Remark 2.2**.**

If $\bm{\beta}$ satisfies (1.10), we denote $C_{\mbox{\tiny{LHLS}}}(\bm{\beta},\bm{\alpha})=\inf_{\bm{\rho}\in\Gamma^{\bm{\beta}}_{2}}\mathcal{F}_{\bm{\alpha}}(\bm{\rho}).$ Obviously, if $\bm{\beta}$ is not critical (i.e., $\Lambda_{I}(\bm{\beta})>0$) then $C_{\mbox{\tiny{LHLS}}}(\bm{\beta},\bm{\alpha})\rightarrow-\infty$ as $\bm{\alpha}\searrow\bm{0}.$

3. Minimizing Movement Scheme

Given two elements $\bm{\rho},\bm{\eta}\in\mathcal{P}^{\bm{\beta}}_{2}(\mathbb{R}^{2}),$ we define the distance between them as follows

[TABLE]

where $d_{\mbox{{\tiny{W}}}}$ is defined by (2.4).

3.1. Minimizing Movement Scheme (MM-scheme):

Given $\bm{\rho}^{0}\in\Gamma^{\bm{\beta}}_{2}$ and a time step $\tau\in(0,1),$ set $\bm{\rho}^{0}_{\tau}=\bm{\rho}^{0}$ and define recursively

[TABLE]

The following proposition vindicates that the MM-scheme (3.1) is well-defined.

Proposition 3.1**.**

• (a)

Assume $\bm{\beta}$ satisfies (1.10) and fix $\bm{\eta}\in\Gamma^{\bm{\beta}}_{2}$ and $\tau>0.$ Then the functional $\mathcal{G}_{\bm{\eta}}$ defined by

[TABLE]

is bounded from below on $\Gamma^{\bm{\beta}}_{2}.$ Moreover, $\mathcal{G}_{\bm{\eta}}$ is sequentially lower semicontinuous with respect to the weak topology in $L^{1}(\mathbb{R}^{2}).$

• (b)

If $\bm{\beta}$ is sub-critical then all the sub-level sets $\{\mathcal{G}_{\bm{\eta}}\leq C\}$ are sequentially precompact with respect to the weak topology in $L^{1}(\mathbb{R}^{2}).$ In particular, the minimization problem $\inf_{\bm{\rho\in\Gamma^{\bm{\beta}}_{2}}}\mathcal{G}_{\bm{\eta}}(\bm{\rho})$ admits a solution.

Proof.

• (a)

The proof of $(a)$ is a simple consequence of Proposition 2.1(b) and the inequality

[TABLE]

which follows from the triangle inequality. As a consequence, we obtain $\mathcal{G}_{\bm{\eta}}(\bm{\rho})\geq\mathcal{F}(\bm{\rho})+\frac{1}{4\tau}M_{2}(\bm{\rho})-\frac{1}{2\tau}M_{2}(\bm{\eta})$ is bounded from below on $\Gamma^{\bm{\beta}}_{2}.$ The sequentially lower semi-continuity of $\mathcal{G}_{\bm{\eta}}$ follows from the sequentially lower semi-continuity of $\mathcal{F}$ (see [SW05, KW]) and that of Wasserstein distance with respect to the weak topology of $L^{1}(\mathbb{R}^{2})$ (see [JKO98]).

• (b)

Any sub-level set $\{\mathcal{G}_{\bm{\eta}}\leq C\}$ is contained in the sub-level set $\{\mathcal{F}_{\bm{\frac{1}{4\tau}}}\leq C+\frac{1}{2\tau}M_{2}(\bm{\eta})\}.$ Therefore, the conclusion of the proposition follows from Proposition 2.1(d) together with the first part of the current proposition.

∎

Remark 3.2**.**

The functional $\mathcal{F}$ is not convex, neither in the usual sense nor it is displacement convex (in the sense of McCann [McC97]). As a consequence, there may not be a unique minimizer in (3.1). We pick a minimizer recursively from the problem (3.1). Surprisingly, every choice gives rise to the same solution to (1.1) in the limit $\tau\rightarrow 0$. Indeed, for the scalar case $n=1,$ Fernández and Mischler [FM16] have obtained the uniqueness of the solutions satisfying the free energy inequality, using an argument introduced by Ben-Artzi [BA94, Bre94] for 2D viscous vortex model. Their argument can be used to the system case and prove the uniqueness of solutions satisfying the free energy inequality.

4. A priori Estimates and the regularity of minimizers

4.1. A priori estimates

Lemma 4.1**.**

For every $T\in(0,\infty)$ there exists a constant $\mathcal{C}_{ap}(T)$ such that for each $\tau\in(0,1)$ and positive integers $k$ satisfying $k\tau\leq T$ there holds

[TABLE]

Proof.

For every $l\in\{1,\ldots,k\},$ the minimizing property of $\bm{\rho}_{\tau}^{l}$ gives

[TABLE]

Summing over $l\in\{1,\ldots,k\}$ we obtain

[TABLE]

Choose $\alpha>0$ such that $8\alpha T<1.$ Applying (3.2), triangle inequality and Cauchy-Schwartz inequality we get

[TABLE]

Adding $\alpha M_{2}(\bm{\rho}_{\tau}^{k})$ on both sides of (4.2), using (4.1) and Proposition 2.1 (b) (and Remark 2.2) we obtain

[TABLE]

which in turn gives

[TABLE]

Again using (4.4) in (4.1) we get

[TABLE]

Finally, since $\mathcal{F}(\bm{\rho}_{\tau}^{k})\leq\mathcal{F}(\bm{\rho}^{0})$ because of (4.2) and $\bm{\beta}$ is sub-critical we obtain an upper bound on the entropy (see [KW, Theorem $4.1$])

[TABLE]

Hence by Lemma A(b) (see appendix) and the bounds (4.5), (4.6) we get

[TABLE]

We conclude the proof with $\mathcal{C}_{ap}(T)=C_{1}(T)+C_{2}(T)+C_{4}(T).$ ∎

4.2. Regularity of minimizers

We indicated in the introduction that in order to derive the Euler-Lagrange equation satisfied by the minimizers in MM-scheme, we need additional regularities on the minimizers. This is the goal of this subsection. We will utilize the flow interchange technique introduced by Matthes-McCann and Savaré [MMS09]. Before stating our regularity results let us first review this technique very briefly in our setting.

4.2.1. Matthes-McCann-Savaré flow interchange technique

Let  $\mathcal{N}:\mathcal{P}^{\bm{\beta}}(\mathbb{R}^{2})\rightarrow\mathbb{R}\cup\{+\infty\}$ be a proper lower semi-continuous functional and let $\mathcal{D}(\mathcal{N}):=\{\bm{\mu}\in\mathcal{P}^{\bm{\beta}}(\mathbb{R}^{2})|\ \mathcal{N}(\bm{\mu})<+\infty\}$ be the domain of $\mathcal{N}.$ Further assume that $\mathcal{N}$ generates a continuous semigroup $\mathcal{S}^{\mathcal{N}}_{t}:\mathcal{D}(\mathcal{N})\rightarrow\mathcal{D}(\mathcal{N}),$ satisfying the evolution variational identity (EVI)

[TABLE]

Assume for simplicity $\Gamma^{\bm{\beta}}_{2}\subset\mathcal{D}(\mathcal{N})$ and $\Gamma^{\bm{\beta}}_{2}$ is an invariant subset under the flow $\mathcal{S}^{\mathcal{N}}_{t}$ i.e., $\mathcal{S}^{\mathcal{N}}_{t}(\Gamma^{\bm{\beta}}_{2})\subset\Gamma^{\bm{\beta}}_{2}.$ If $\bm{\mu}\in\Gamma^{\bm{\beta}}_{2}$ we define the dissipation of $\mathcal{F}$ along the flow $(\mathcal{S}^{\mathcal{N}}_{t})_{t\geq 0}$ of $\mathcal{N}$ by

[TABLE]

By the minimizing property of $\bm{\rho}_{\tau}^{k}$ (see (3.1))

[TABLE]

for all $\bm{\rho}\in\Gamma^{\bm{\beta}}_{2}.$ Now setting $\bm{\rho}=\mathcal{S}^{\mathcal{N}}_{t}(\bm{\rho}_{\tau}^{k})$ in (4.9) for $t>0,$ dividing by $t$ and using (4.8) we get

[TABLE]

Since $\mathcal{N}$ is lower semi-continuous, passing to the limit as $t\searrow 0$ we obtain

[TABLE]

In addition, if the quantity $\bm{\nabla}^{\mathcal{N}}\mathcal{F}\geq 0,$ or, at least behaves nicely then we could possibly get adequate estimates on the minimizers $\bm{\rho}_{\tau}^{k}.$

Remark 4.2**.**

The dissipation of $\mathcal{F}$ along the flow $\mathcal{S}^{\mathcal{N}}_{t}$ was originally denoted by $\bm{D}^{\mathcal{N}}$ in the article [MMS09]. In order to avoid confusion with the domain of $\mathcal{N}$ we choose to denote it by $\bm{\nabla}^{\mathcal{N}}.$

In our case we use the entropy functional

[TABLE]

if $\bm{\rho}$ is absolutely continuous with respect to the Lebesgue measure and $+\infty$ everywhere else. The functional $\mathcal{H}$ is a particular example of the class known as displacement convex entropy, which guarantees the existence of a continuous semigroup [AGS05]. However, we don’t have $\bm{\nabla}^{\mathcal{H}}\mathcal{F}\geq 0$, but we can control the negative part of $\bm{\nabla}^{\mathcal{H}}\mathcal{F}$ and achieve higher regularity of the minimizers. Below we recall some few well-known facts on the displacement convex entropy in $\mathbb{R}^{2}.$

4.2.2. Displacement convex entropy

Let $U:[0,+\infty)\rightarrow\mathbb{R}$ be a convex function satisfying

• •

$U(0)=0,$ $U$ is continuous at [math] and $U\in C^{1}(0,\infty),$

• •

$\lim_{t\rightarrow+\infty}\frac{U(t)}{t}=+\infty$ and $\lim_{t\rightarrow 0+}\frac{U(t)}{t^{\alpha}}>-\infty$ for some $\alpha>\frac{1}{2},$

• •

$t\mapsto t^{2}U(t^{-2})$ is convex and decreasing in $(0,\infty).$

Define the functional $\mathcal{U}:\mathcal{P}_{2}(\mathbb{R}^{2})\rightarrow\mathbb{R}\cup\{+\infty\}$ by

[TABLE]

The domain of $\mathcal{U}$ denoted by $\mathcal{D}(\mathcal{U})$ is the set of all densities $\rho$ such that $\mathcal{U}(\rho)<+\infty.$ Such a functional is called a displacement convex entropy with density function $U.$ It is well known that a displacement convex entropy $\mathcal{U}$ generates a continuous semigroup $\mathcal{S}^{\mathcal{U}}_{t}:\mathcal{D}(\mathcal{U})\rightarrow\mathcal{D}(\mathcal{U})$ satisfying the Evolution Variational Identity

[TABLE]

and $w:=\mathcal{S}^{\mathcal{U}}_{t}(\rho)$ is the unique distributional solution to the Cauchy problem

[TABLE]

where $L_{U}(t)=tU^{\prime}(t)-U(t).$ We refer to [AGS05, Theorem $11.2.5$] for a proof of the claims described above.

It is easy to see that $H(t):=t\ln t$ satisfies above all criterion of a displacement convex entropy. Also, note that for our purpose we need to apply the above results to the densities $\rho$ having mass $\beta>0.$ Let $\rho,\eta$ be two densities with mass $\beta$ having finite entropies and finite second moments. Then applying EVI to the normalized densities gives

[TABLE]

since $L_{H}(t)=t,$ and $w:=\mathcal{S}^{\mathcal{H}}_{t}(\rho)$ is a distributional solution to

[TABLE]

Moreover, note that $M_{2}(\mathcal{S}^{\mathcal{H}}_{t}(\rho))<+\infty$ for all $t$ since $M_{2}(\rho)<+\infty.$

4.2.3. Regularity results

Lemma 4.3**.**

Let $\tau\in(0,1)$ and let $\bm{\rho}_{\tau}^{k}$ be a sequence obtained using the MM-scheme (3.1) satisfying

[TABLE]

for some constant $\Theta>0.$ Then $\rho_{\tau,i}^{k}\in W^{1,1}(\mathbb{R}^{2}),\frac{\nabla\rho_{\tau,i}^{k}}{\rho_{\tau,i}^{k}}\in L^{2}(\mathbb{R}^{2},\rho_{\tau,i}^{k})$ for all $i\in I.$

Moreover, there exists a constant $C(\Theta)$ such that

[TABLE]

Proof.

For each $i\in\{1,\ldots,n\},$ let $\mathcal{S}_{t}^{\mathcal{H}}:\mathcal{D}(\mathcal{H})\rightarrow\mathcal{D}(\mathcal{H})$ be the continuous semigroup generated by $\mathcal{H}$ with respect to the Wasserstein distance in $\mathcal{P}^{\beta_{i}}_{2}(\mathbb{R}^{2})$. For simplicity of notations we define $w_{t,i}=\mathcal{S}^{\mathcal{H}}_{t}(\rho_{\tau,i}^{k})$ and $\bm{w}_{t}=(w_{t,1},\ldots,w_{t,n}).$ Then $w_{t,i}$ satisfies

[TABLE]

Moreover, since $\bm{\rho}_{\tau}^{k}\in\Gamma^{\bm{\beta}}_{2}$ we get the following outcomes: for all $i\in\{1,\ldots,n\}$

• (a)

for each $T>0$ there exists a constant $C(T)>0$ such that $M_{2}(w_{t,i})\leq C(T),$ for all $t\leq T,$

• (b)

$\lim_{t\rightarrow 0+}||w_{t,i}-\rho_{\tau,i}^{k}||_{L^{1}(\mathbb{R}^{2})}=0,$

• (c)

$\lim_{t\rightarrow 0+}\int_{\mathbb{R}^{2}}w_{t,i}\ln w_{t,i}\ dx=\int_{\mathbb{R}^{2}}\rho_{\tau,i}^{k}\ln\rho_{\tau,i}^{k}\ dx.$

• (d)

$\frac{d}{dt}\int_{\mathbb{R}^{2}}w_{t,i}\ln w_{t,i}\ dx=-\int_{\mathbb{R}^{2}}\left|\frac{\nabla w_{t,i}}{w_{t,i}}\right|^{2}w_{t,i}\ dx<0,$ for $t>0.$

For the convenience of the reader we prove $(c)$. Since $H$ is convex by strong $L^{1}$-convergence $(b)$ ($L^{1}$-weak convergence is enough though) we have

[TABLE]

On the other hand, by uniqueness of the solutions to the heat equation, $w_{t,i}$ is the convolution (with respect to the space variable) of $\rho_{\tau,i}^{k}$ with the heat kernel $G_{t}(x):=\frac{1}{\sqrt{4\pi t}}e^{-\frac{|x|^{2}}{4t}}.$ Again by convexity of $H$ and Jensen’s inequality

[TABLE]

Integrating with respect to $x$ and taking $\limsup$ as $t\rightarrow 0+$ we obtain

[TABLE]

Step 1. $t\mapsto\mathcal{F}(\bm{w}_{t})$ is continuous at [math] and differentiable in $(0,\infty).$

Proof of step 1. The differentiability property of the functional follows from the smoothness of $w_{t}$ as it is being the convolution of $\rho_{\tau,i}^{k}$ with the heat kernel $G_{t}(x).$ So, we only need to check the continuity at $0.$ In fact, by $(a),(b)$ and $(c)$ we can argue as in [BCC08, Lemma $3.1$] (see also [KW]) to obtain

[TABLE]

and hence $\lim_{t\rightarrow 0+}\mathcal{F}(\bm{w}_{t})=\mathcal{F}(\bm{\rho}_{\tau}^{k}).$

Define $u_{t,i}(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\ln|x-y|w_{t,i}(y)\ dy,$ then $-\Delta u_{t,i}=w_{t,i}.$ Owing to the smoothness of $\bm{w}_{t}$ and using the symmetry of $(a_{ij})$ we get for $t>0$

[TABLE]

Step 2. There exists a constant $C_{1}(\Theta)$ such that

[TABLE]

Proof of step 2. By Cauchy-Schwartz inequality, Lemma A(a) (see appendix) and $(d)$

[TABLE]

Choosing $\epsilon>0$ small such that $C_{3}(\Theta)\epsilon<\frac{1}{2}$ and using (4.2.3) we obtain (4.14).

Step 3: $L^{2}$-regularity of $\bm{\rho}_{\tau}^{k}.$

Proof of step 3. Let $t(m)$ be a sequence such that $t(m)\searrow 0$ as $m\rightarrow\infty.$ Since $t\mapsto\mathcal{F}(w_{t})$ is continuous on $[0,\infty)$ and differentiable in $(0,\infty),$ by Lagrange mean value theorem there exists $r(m)\in(0,t(m))$ such that

[TABLE]

Combining (4.10) and (4.15) we get

[TABLE]

Now Step 2 and (4.16) gives

[TABLE]

Since $\mathcal{H}(\bm{w}_{t})$ remains bounded for small time (thanks to $(d)$) we get from (4.17)

[TABLE]

Using Lemma A$(a)$ again, we obtain

[TABLE]

Since $L^{2}(\mathbb{R}^{2})$ is a reflexive Banach space then, up to a subsequence (not labeled) we get $w_{r(m),i}\rightharpoonup w_{i}$ weakly in $L^{2}(\mathbb{R}^{2})$ as $m\rightarrow\infty.$ But we already know that $w_{r(m),i}\rightarrow\rho_{\tau,i}^{k}$ strongly in $L^{1}(\mathbb{R}^{2})$ as $m\rightarrow\infty.$ Hence, we must have $\rho_{\tau,i}^{k}=w_{i},$ proving $\rho_{\tau,i}^{k}\in L^{2}(\mathbb{R}^{2}).$

Step 4: $W^{1,1}$-regularity of $\bm{\rho}_{\tau}^{k}$ and the proof of (4.12).

Proof of step 4. Define $v_{m,i}:=\frac{\nabla w_{r(m),i}}{w_{r(m),i}}.$ Since $w_{r(m),i}$ narrowly converges to $\rho_{\tau,i}^{k}$ as $m\rightarrow\infty$ and (4.18) holds, we can invoke Proposition A to conclude that there exists a vector field $v_{i}\in L^{2}(\mathbb{R}^{2},\rho_{\tau,i}^{k};\mathbb{R}^{2})$ such that

[TABLE]

for all $\zeta\in C_{c}^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2})$ and $i=1,\ldots,n$ and moreover,

[TABLE]

Now since $v_{i}\rho_{\tau,i}^{k}\in L^{1}(\mathbb{R}^{2})\times L^{1}(\mathbb{R}^{2})$ and

[TABLE]

we conclude from (4.19) and (4.21) that $\nabla\rho_{\tau,i}^{k}=v_{i}\rho_{\tau,i}^{k}.$ Which proves $\rho_{\tau,i}^{k}\in W^{1,1}(\mathbb{R}^{2}).$ Finally from (4.20), (4.17) and $(c)$ we get

[TABLE]

This completes the proof of the lemma. ∎

Next we obtain regularity estimates on the Newtonian potential of $\rho_{\tau,i}^{k}.$ Recall that the Newtonian potential of $\rho_{\tau,i}^{k}$ is defined by

[TABLE]

Lemma 4.4**.**

Let $\tau\in(0,1)$ be given and assume that $\bm{\rho}_{\tau}^{k}$ satisfies (4.11). Let $u_{\tau,i}^{k}$ be the Newtonian potential defined by (4.22). Then for $i\in\{1,\ldots,n\}$

• (a)

there exists a constant $C_{\Theta}>0$ and $R_{0}>0$ such that

[TABLE]

• (b)

For each $R>0,$ there exists a constant $C_{R,\Theta}>0$ such that

[TABLE]

Proof.

$(a).$ Follows from the results of Chen-Li [CL93] and by the hypothesis (4.11), see [KW, Lemma $3.5$] for details.

$(b).$ For the first part, note that by $(a),$ $|u_{\tau,i}^{k}(x)|\leq C_{1}(\Theta,R)$ for $x\in\partial B_{3R}$ and for some constant $C_{1}(\Theta,R)>0$ depending only on $\Theta$ and $R.$ Define $v_{i}^{k}$ as follows

[TABLE]

Then by assumption (4.11) and invoking the results of [Sta63, AF95] we obtain $v_{i}^{k}\in H^{1}_{0}(B_{4R})\cap L^{\infty}(B_{4R})$ and $||v_{i}^{k}||_{L^{\infty}(B_{4R})}+||v_{i}^{k}||_{H^{1}(B_{4R})}\leq C_{2}(\Theta,R).$ Now, $w_{i}^{k}:=u_{\tau,i}^{k}-v_{i}^{k}$ is Harmonic in $B_{4R}$ and satisfies $|w_{i}^{k}(x)|\leq C_{3}(\Theta,R)$ for all $x\in\partial B_{3R}.$ By maximum principle, $|w_{i}^{k}(x)|\leq C_{3}(\Theta,R),$ for all $x\in B_{3R}.$ Furthermore, since $w_{i}^{k}$ is harmonic

[TABLE]

where $\alpha$ is a multi-index and $|\alpha|$ denotes the sum of all its components. As a result, $||w_{i}^{k}||_{H^{1}(B_{R})}\leq C_{5}(\Theta,R)$ and hence

[TABLE]

For the second part, on one hand, by Lemma A(a), Lemma 4.3 and by assumption on $\rho_{\tau,i}^{k}$ (equation (4.11))

[TABLE]

So by elliptic regularity, $v_{i}^{k}$ defined in (4.23) satisfies

[TABLE]

On the other hand by derivative estimate (4.24), $||w_{i}^{k}||_{H^{2}(B_{R})}\leq C_{8}(\Theta,R).$ Choosing appropriate $\epsilon,$ we conclude the proof. ∎

5. The Euler Lagrange equation

Lemma 5.1**.**

Let $\bm{\beta}$ be sub-critical, $\bm{\rho}^{0}\in\Gamma^{\bm{\beta}}_{2}$ and $\tau>0$ be given. Let $\bm{\rho}_{\tau}^{k}$ be the sequence obtained iteratively by the MM-scheme (3.1). If $\nabla\varphi^{k}_{i}$ denotes the map transporting $\rho_{\tau,i}^{k-1}$ to $\rho_{\tau,i}^{k}$ then

• (a)

Euler-Lagrange equation:

[TABLE]

holds for all $i=1,\ldots,n$ and any $\zeta\in C_{c}^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2}).$

• (b)

Free energy production term: the following identity holds

[TABLE]

• (c)

The following approximate weak solution is satisfied

[TABLE]

for all $\psi\in C_{c}^{\infty}(\mathbb{R}^{2}).$

Proof.

(a) Let $\zeta_{i}\in C_{c}^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2}),i=1,\ldots,n$ be $n$-smooth vector fields. For $\epsilon>0$ and for each $i$ define $T_{\epsilon,i}:=x+\epsilon\zeta_{i}.$ Then for $\epsilon>0$ small enough $\det\nabla T_{\epsilon,i}=\det(I+\epsilon\nabla\zeta_{i})>0,$ and so $T_{\epsilon,i}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is a $C^{\infty}$-diffeomorphism. Let $\rho_{\epsilon,i}$ be the push forward of $\rho_{\tau,i}^{k}$ under the map $T_{\epsilon,i}$ (i.e., $\rho_{\epsilon,i}=T_{\epsilon,i}\#\rho_{\tau,i}^{k}$) and let $\bm{\rho}_{\epsilon}=(\rho_{\epsilon,1},\ldots,\rho_{\epsilon,n}).$ Then by change of variable formula (2.5)

[TABLE]

Since $\nabla\varphi^{k}_{i}\#\rho_{\tau,i}^{k-1}=\rho_{\tau,i}^{k}$ and $T_{\epsilon,i}\#\rho_{\tau,i}^{k}=\rho_{\epsilon,i}$ the map $T_{\epsilon,i}\circ\nabla\varphi^{k}_{i}$ transports $\rho_{\tau,i}^{k-1}$ onto $\rho_{\epsilon,i}$. By definition of the Wasserstein distance

[TABLE]

Using the minimizing property of $\bm{\rho}_{\tau}^{k},$ (5) and (5.5) we get

[TABLE]

Dividing by $\epsilon>0$ and letting $\epsilon\rightarrow 0$ we get

[TABLE]

Finally, fixing an $i$ and choosing $\zeta_{i}=\pm\zeta\in C_{c}^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2})$ and $\zeta_{j}=0$ for all $j\neq i$ we obtain

[TABLE]

Recall that $\nabla u^{k}_{\tau,i}(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\frac{x-y}{|x-y|^{2}}\rho_{\tau,i}^{k}(y)\ dy.$ By regularity results of Lemma 4.3, Lemma 4.4, $\rho_{\tau,i}^{k}\in L^{2}(\mathbb{R}^{2})$ and $|\nabla u^{k}_{\tau,i}|\in L^{2}_{loc}(\mathbb{R}^{2}).$ Therefore, the last term in (5) makes sense.

Using the definition of $\nabla u^{k}_{\tau,j}$ in (5) and applying integration by parts on the second term we deduce ((a)).

(b) Let $\varphi^{k,*}_{i}$ denotes the convex conjugate of $\varphi^{k}_{i}.$ Since both $\rho_{\tau,i}^{k},\rho_{\tau,i}^{k-1}\in\Gamma^{\bm{\beta}}_{2},$ in particular, absolutely continuous with respect to the Lebesgue measure, $\nabla\varphi^{k,*}_{i}\#\rho_{\tau,i}^{k}=\rho_{\tau,i}^{k-1}.$ Using this we can rewrite ((a)) as

[TABLE]

Since this is true for any $\zeta\in C_{c}^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2})$ we conclude

[TABLE]

Recalling the definition of Wasserstein distance and (5.8) we get

[TABLE]

(c) Finally, for $\psi\in C_{c}^{\infty}(\mathbb{R}^{2})$ putting $\zeta=\nabla\psi$ in ((a)) and using the Taylor expansion formula

[TABLE]

we can rewrite the left hand side of ((a)) as

[TABLE]

Inserting the last identity in ((a)) and multiplying by $\tau$ we obtain ((c)). ∎

6. Estimates on the time interpolation

6.1. Time interpolation

We define the piecewise constant time dependent interpolation

[TABLE]

In the subsequent sections, we will show that for any time $T>0,$ the piecewise constant interpolates $\bm{\rho}_{\tau}$ converges in some sense to a solution (according to definition (1.2)) to the PKS-system (1.1) satisfying the energy dissipation inequality (or, free energy inequality).

Lemma 6.1**.**

For every $T>0,$ there exists a constant $\mathcal{C}_{in}(T)>0$ such that for every $\tau\in(0,1)$

[TABLE]

Moreover, the Newtonian potentials $u_{\tau,i}(t),t\in(0,T)$ satisfy the uniform log decay and locally uniform $H^{1}$-estimate stated in Lemma 4.4. In addition, for every $R>0$ there exists a constant $\mathcal{C}(R,T)$ independent of $\tau$ such that

[TABLE]

Proof.

Note that by a priori estimate Lemma 4.1, there exists a constant $\mathcal{C}_{ap}(T)$ such that for all $1\leq l\leq k$ satisfying $k\tau\leq T$ we have

[TABLE]

Applying Lemma 4.3, with $\Theta=\mathcal{C}_{ap}(T)$ we get

[TABLE]

for some constant $C(T)$ depending on $\mathcal{C}_{ap}(T)$ and for all $l=1,\ldots,k.$ Summing over $l$ and multiplying by $\tau$ gives

[TABLE]

Using the definition of $\bm{\rho}_{\tau}$ and again using Lemma 4.1 we get

[TABLE]

The $L^{2}((0,T)\times\mathbb{R}^{2})$ estimate follows from Lemma A(a) and (6.1). Finally, by definition $u_{\tau,i}(t)=u_{\tau,i}^{k}$ if $t\in((k-1)\tau,k\tau].$ Therefore, the conclusion of the second part of the lemma follows from Lemma 4.4 and proceeding as the proof of (6.1). ∎

Lemma 6.2**.**

Let $T>0$ be given. There exists a constant $\mathcal{C}_{con}(T)>0$ such that for all $(s,t)\in[0,T]\times[0,T]$ and $\tau\in(0,1)$

[TABLE]

Proof.

With out loss of generality we can assume that $0\leq s<t.$ Let $k=[t/\tau]+1$ and $\tilde{k}=[s/\tau]+1$ (where $[a]$ denotes the largest integer smaller than $a$). Then using the definition of $\bm{\rho}_{\tau}$ and Lemma 4.1 we get

[TABLE]

∎

7. Convergence of the MM scheme

We are going to apply the refined Arzelà-Ascoli Theorem A (see appendix) to the following choices: $\mathcal{S}=\mathcal{P}_{2}^{\beta_{i}}(\mathbb{R}^{2}),$ $\sigma$ = narrow convergence in $\mathcal{P}_{2}^{\beta_{i}}(\mathbb{R}^{2}),$ $K=\{\rho_{\tau,i}(t)|\ t\in[0,T],\ \tau\in(0,1)\},$ $d=d_{\mbox{{\tiny{W}}}},$ $w(s,t)=\sqrt{|s-t|},$ note that here $\mathcal{B}=\emptyset.$

7.1. Convergence of the time interpolation

By Lemma 4.1

[TABLE]

Therefore the set $\{\rho_{\tau,i}(t)|\ t\in[0,T],\ \tau\in(0,1)\}$ is weakly sequentially compact in $L^{1}(\mathbb{R}^{2})$ (in particular, sequentially compact with respect to $\sigma$). Moreover, by Lemma 6.2

[TABLE]

Therefore by Theorem A, there exists a curve $\bm{\rho}(t):[0,T]\rightarrow(L^{1}(\mathbb{R}^{2}))^{n}$ and a monotone decreasing sequence $\tau_{m}\rightarrow 0$ such that

• •

$\rho_{\tau_{m},i}(t)\rightharpoonup\rho_{i}(t)$ weakly in $L^{1}(\mathbb{R}^{2})$ for every $t\in[0,T].$

• •

$\rho_{i}$ is $d_{\mbox{{\tiny{W}}}}$ continuous. More precisely, $d_{\mbox{{\tiny{W}}}}(\rho_{i}(t),\rho_{i}(s))\leq\mathcal{C}_{con}(T)\sqrt{|t-s|}.$ Thus $\rho_{i}\in C^{0,\frac{1}{2}}([0,T];\mathcal{P}_{2}^{\beta_{i}}(\mathbb{R}^{2}))$ for $i=1,\ldots,n.$

Furthermore, by Lemma 6.1, $\rho_{\tau_{m},i}$ is bounded in $L^{2}((0,T)\times\mathbb{R}^{2})$ and hence

[TABLE]

proving that $\rho_{i}\in L^{2}((0,T)\times\mathbb{R}^{2}).$ Finally, consider the sequences

[TABLE]

By Lemma 6.1 and Proposition A, applied to the probability measures $d\mu_{m}$ and the vector fields $\tilde{v}_{m}$ we see that there exists a vector field $v\in L^{2}((0,T)\times\mathbb{R}^{2},\rho_{i}\ ;\mathbb{R}^{2})$ such that

[TABLE]

Moreover, since $v\rho_{i}\in L^{1}((0,T)\times\mathbb{R}^{2};\mathbb{R}^{2})$ and

[TABLE]

we conclude $v\rho_{i}=\nabla_{x}\rho_{i}$ and hence $\rho_{i}\in L^{1}((0,T);W^{1,1}(\mathbb{R}^{2})).$ By lower semicontinuity (Proposition A equation (9.2)) and Lemma 6.1

[TABLE]

proving that $\frac{\nabla\rho_{i}}{\rho_{i}}\in L^{2}((0,T)\times\mathbb{R}^{2},\rho_{i}\ ;\mathbb{R}^{2})$ for all $i\in\{1,\ldots,n\}.$

7.2. Convergence of the Newtonian potential

This is a crucial step because apparently we do not have explicit continuity in time estimates on $\bm{u}_{\tau_{m}}$ as in Lemma 6.2 for $\bm{\rho}_{\tau_{m}}$. Fortunately, we do have continuity in time estimate on the harmonic part of $\bm{u}_{\tau_{m}},$ which together with the results of section $7.1$ enables us to prove the following lemma: set

[TABLE]

Lemma 7.1**.**

Up to a subsequence $u_{\tau_{m},i}(t)\rightarrow u_{i}(t)$ strongly in $L^{2}(0,T;L^{2}(\mathbb{R}^{2},\omega))$.

Proof.

Since $\rho_{i}\in L^{2}((0,T)\times\mathbb{R}^{2})),$ for almost every $t\in[0,T],\ ||\rho_{i}(t)||_{L^{2}(\mathbb{R}^{2})}<+\infty.$ Denote $\iota:=\{t\in[0,T]|\ ||\rho_{i}(t)||_{L^{2}(\mathbb{R}^{2})}=+\infty\}.$ By Lemma 6.1, for each $t\in[0,T],\ u_{\tau_{m},i}(t)$ is bounded in $H^{1}_{loc}(\mathbb{R}^{2}).$ Hence we can extract a countable dense subset $\mathcal{K}$ of $[0,T]\backslash\iota$ and a subsequence (indexed by $\tau_{m}$ itself) such that

[TABLE]

Since, by Lemma 6.1, $u_{\tau_{m},i}$ has uniform log decay we deduce

[TABLE]

Next we need some continuity in time estimates on $u_{\tau_{m},i}$ in order to conclude the above $L^{2}$-convergence holds for all $t\in[0,T]\backslash\iota.$

Continuity estimate 1: by Lemma 6.2, for every $s,t\in[0,T]$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{2})$

[TABLE]

Continuity estimate 2: the quadratic interaction term is continuous with respect to the weak $L^{1}$-convergence. Particularly, for each $t\in[0,T],$

[TABLE]

For a proof of (7.4) see [BCC08, Lemma $2.3$ and the proof of Lemma $3.1$]. Here the crucial point is that $\rho_{\tau_{m},i}(t)\rightharpoonup\rho_{i}(t)$ weakly in $L^{1}(\mathbb{R}^{2})$ for all $t\in[0,T]$ and has uniform entropy and second moment bound.

Now take any $s\in[0,T]\backslash(\mathcal{K}\cup\iota).$ Let $u_{\tau_{m}^{\prime},i}(s)$ be a subsequence of $u_{\tau_{m},i}(s)$ such that $u_{\tau_{m}^{\prime},i}(s)\rightarrow v_{i}(s)$ strongly in $L^{2}_{loc}(\mathbb{R}^{2})$ for some $v_{i}(s)\in H^{1}_{loc}(\mathbb{R}^{2}).$ Along the same subsequence passing to the limit in (7.2) we get

[TABLE]

This, and $\rho_{i}\in C^{0,\frac{1}{2}}([0,T];\mathcal{P}_{2}^{\beta_{i}}(\mathbb{R}^{2}))$ imply that

[TABLE]

Hence $u_{i}(s)-v_{i}(s)$ is Harmonic in $\mathbb{R}^{2}.$ But both have log decay and therefore $u_{i}(s)-v_{i}(s)$ must be a constant. Denote this constant by $\varepsilon_{i}.$ We claim that $\varepsilon_{i}=0.$ Indeed, by strong $L^{2}$-convergence of $u_{\tau_{m}^{\prime},i}(s)$ and using $\rho_{i}(s)\in L^{2}(\mathbb{R}^{2})$ we obtain

[TABLE]

On the other hand by (7.4)

[TABLE]

Hence (7.6) and (7.7) we get $\epsilon_{i}=0.$ By Lemma 6.1, $||u_{\tau_{m},i}(t)||_{L^{2}(\mathbb{R}^{2},\omega)}^{2}\leq C(T)$ for all $t\in[0,T].$ We can apply dominated convergence theorem to conclude the Lemma. ∎

Let $\xi\in C_{c}^{\infty}([0,T)\times\mathbb{R}^{2})$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{2})$ be a smooth compactly supported function such that $\psi=1$ on a domain containing the support of $\xi(t)$ for all $t.$ Then again by Lemma 6.1, the sequence $\{\psi u_{\tau_{m},i}\}$ is bounded in $L^{2}(0,T;H^{2}(\mathbb{R}^{2})).$ By Lemma 7.1, Lemma B (applied to $X=H^{2}(\mathbb{R}^{2}),B=H^{1}(\mathbb{R}^{2},\omega),Y=L^{2}(\mathbb{R}^{2},\omega),F=\{\psi u_{\tau_{m},i}\}$) we deduce

[TABLE]

7.3. Rewriting the Euler-Lagrange equation

Set $N_{m}:=[T/\tau_{m}]$ and take $\xi$ of class $C_{c}^{\infty}([0,T)\times\mathbb{R}^{2}).$ Then

[TABLE]

Since $\xi\equiv 0$ at $t=T$

[TABLE]

where we used $|T-N_{m}\tau_{m}|\leq\tau_{m}.$ For each $k\in\{1,\ldots,N_{m}\}$ applying ((c))(Lemma 5.1(c)) with $\psi=\xi((k-1)\tau_{m})$ we get

[TABLE]

Summing over $k=1,\ldots,N_{m}$ and using Lemma 4.1 we get

[TABLE]

We write $(I_{1})+(I_{2})+(I_{3})$ for the terms on the left hand side of (7.3). Thanks to (7.3) and (7.10) we can write

[TABLE]

For $(I_{2})$ and $(I_{3})$ we use the following estimates:

[TABLE]

where we have used $||\nabla\rho_{\tau_{m},i}^{k}||_{L^{1}(\mathbb{R}^{2})}\leq\sqrt{\beta_{i}}||\nabla\rho_{\tau_{m},i}^{k}/\rho_{\tau_{m},i}^{k}||_{L^{2}(\mathbb{R}^{2},\rho_{\tau_{m},i}^{k})}=O(1/\sqrt{\tau_{m}}),$ thanks to Lemma 4.3. Similarly,

[TABLE]

By Cauchy-Schwartz and Lemma 4.4

[TABLE]

Therefore from (7.3)-(7.3) combined with the three estimates in (7.12) we get

[TABLE]

7.4. Passing to the limit

For the first term in (7.3) we use $\rho_{\tau_{m},i}\rightharpoonup\rho_{i}$ in $L^{2}((0,T)\times\mathbb{R}^{2}),$ for the third term we use (7.1), and finally for the last term we use (7.8), the weak convergence of $\rho_{\tau_{m},i}$ to $\rho_{i}$ in $L^{2}((0,T)\times\mathbb{R}^{2})$ and duality to pass to the limit. Therefore we conclude $\rho_{i}$ satisfies

[TABLE]

for all $i=1,\ldots,n$ and any test function $\xi\in C_{c}^{\infty}([0,T)\times\mathbb{R}^{2}).$ Which is the weak formulation of the PKS-system (1.1) (see Definition 1.2).

8. Free energy inequality

In this section, we show, by using De Giorgi variational interpolation, that the obtained solution satisfies the free energy inequality. Define for $\tau\in(0,1)$

[TABLE]

With out loss of generality we can assume that $\bm{\tilde{\rho}}_{\tau}(k\tau)=\bm{\rho}_{\tau}^{k}.$

Remark 8.1**.**

One can proceed as in Lemma 4.1 and obtain local uniform entropy and moment bound on $\bm{\tilde{\rho}}_{\tau}(t).$ As a consequence, for each $t,\ \tilde{\rho}_{\tau,i}(t)$ exhibits the same regularity properties stated in the first part of the Lemma 4.3. It is not, however, clear that $\bm{\tilde{\rho}}_{\tau}(t)$ has finite Fisher information bound, which is essential to obtain the $H^{2}$-regularity estimate on the Newtonian potential of $\bm{\tilde{\rho}}_{\tau}(t)$ stated in Lemma 6.1.

The following discrete energy identity serves the purpose in this regard and it is also an imperative step to prove the free energy inequality.

Lemma 8.2** (Discrete energy identity).**

For every $k\in\mathbb{N}$ and $\tau\in(0,1)$ the De-Giorgi interpolation defined by (8.1) satisfies the following energy identity:

[TABLE]

Furthermore, for every $T>0$ there exists a constant $C(T)>0$ such that

[TABLE]

Proof.

Proceeding exactly as in [AGS05, Theorem $3.1.4$ and Lemma $3.2.2$] we have

[TABLE]

Summing over all $l\in\{1,\ldots,k\}$ we deduce

[TABLE]

where

[TABLE]

Recall that by Lemma 5.1(b)

[TABLE]

Applying the same argument of Lemma 5.1 to $\bm{\tilde{\rho}}_{\tau}$ we infer that

[TABLE]

for all $t\in((l-1)\tau,l\tau].$ Plugging (8.4) and (8.5) into (8.3) and using the definition of $\bm{\rho}_{\tau}$ we obtain the aforementioned discrete energy identity. The proof of (8.2) is similar to the proof of [AGS05, Lemma $3.2.2$ equation $(3.2.7)$]. ∎

Lemma 8.3**.**

For every $T\in(0,\infty)$ there exists a constant $\mathcal{C}_{de}(T)>0$ such that for every $\tau\in(0,1),$ the De Giorgi interpolates defined by (8.1) satisfies

[TABLE]

Proof.

By Lemma 8.2 and Lemma 4.1 we get

[TABLE]

for some constant $C(T)>0$ which implicitly depends on $\mathcal{C}_{ap}(T).$ Now can write for an $t\in[0,T]$

[TABLE]

Next we make use of the following inequality whose proof can be found in [FHM14, Lemma $3.2$](see also [FM16, Lemma $2.1$]): There exists a constant $C_{p}>0$ such that

[TABLE]

holds true for all $p\in[1,\infty)$ and for all $f\in L^{1}_{+}(\mathbb{R}^{2})$ satisfying $\int_{\mathbb{R}^{2}}\frac{|\nabla f|^{2}}{f}\ dx<\infty.$

Choose $M>1$ a large number whose value will be decided later. An application of Hölder inequality and (8.8) with $p=3$ gives, as in [FHM14]

[TABLE]

where $\mathbbm{1}_{A}$ denotes the characteristic function of a set $A.$ Choosing $M$ large and utilizing (8) and the entropy bound (see Remark 8.1) we can estimate

[TABLE]

On the other hand

[TABLE]

Combining (8),(8.10) and (8.11) we deduce for every $t\in[0,T]$

[TABLE]

Integrating with respect to $t$ from [math] to $T$ and using (8.6) we conclude the proof. ∎

Finally, we are in a position to prove the free energy inequality.

Proof of Theorem 1.3$(b)$:

Proof.

Fix $T>0$ and choose the sequence $\tau_{m}$ as in section $7$ so that $\rho_{\tau_{m},i}(t)\rightharpoonup\rho_{i}(t)$ weakly in $L^{1}(\mathbb{R}^{2})$ for all $t\in[0,T]$ and $\rho_{\tau_{m},i}\rightharpoonup\rho_{i}$ weakly in $L^{2}((0,T)\times\mathbb{R}^{2}).$ Moreover, the Newtonian potentials $u_{\tau_{m},i}\rightarrow u_{i}$ strongly in $L^{2}((0,T);H^{1}_{loc}(\mathbb{R}^{2})).$

Furthermore by Lemma 8.2, Lemma 8.3 and Remark 8.1, the De-Giorgi interpolation $\bm{\tilde{\rho}}_{\tau_{m}}$ enjoys the same property and also converges to the same limit $\bm{\rho}.$ Set

[TABLE]

$\tilde{v}_{m,i}=(v_{m,i},1)$ and $d\mu_{m,i}=\frac{1}{T\beta_{i}}\rho_{\tau_{m},i}dxdt.$ Note that by Lemma 8.2 and Lemma 4.1

[TABLE]

Invoking Proposition A again we obtain the existence a vector field $v_{i}\in L^{2}((0,T)\times\mathbb{R}^{2},\rho_{i};\mathbb{R}^{2})$ such that

[TABLE]

Now proceeding as the proof of (7.1) goes, we derive

[TABLE]

Since $v_{i}\rho_{i}\in L^{1}((0,T)\times\mathbb{R}^{2})$ we conclude $v_{i}\rho_{i}=\nabla\rho_{i}-\sum_{j=1}^{n}a_{ij}\nabla u_{j}\rho_{i}$ and moreover, by lower semicontinuity (Proposition A (9.2))

[TABLE]

holds. Similarly, (8) holds for $\bm{\tilde{\rho}}_{\tau_{m}}.$ Finally, passing to the limit in the discrete energy identity (Lemma 8.2) and using the lower semi-continuity of $\mathcal{F}$ with respect to the narrow convergence we get

[TABLE]

This completes the proof of Theorem 1.3(b). ∎

Proof of Theorem 1.3$(c)$:

Proof.

The proof of uniqueness follows directly as in the scalar case ($n=1$) [FM16], so we only sketch the main steps. First, prove the hypercontractivity result [Gro75, BDP06] for the weak solution, using a variant of Diperna-Lions renormalizing trick in spirit of [FHM14] and [FM16], extended for the system. For this we apply the a-posteriori estimate (7.2) from part (b) of the Theorem. As a result we get that any weak solution with finite initial entropy is smooth at $t>0$. Then we get the estimate

[TABLE]

Next we consider two such weak solutions $\rho_{i}^{0},\rho_{i}^{1}$ which agrees at $t=0$, and let $Q(t)=\sup_{0<s\leq t}\sum_{i=1}^{n}s^{\frac{1}{4}}||F_{i}(s)||_{L^{\frac{4}{3}}(\mathbb{R}^{2})}$ where $F_{i}(s):=\rho^{0}_{i}(\cdot,s)-\rho^{1}_{i}(\cdot,s)$. Then we get the estimate for $Q(t)$ via the heat kernel

[TABLE]

applied to $\rho_{i}^{0}$ and $\rho_{i}^{1}$ to obtain

[TABLE]

where $P^{0},P^{1}$ stands for $\rho^{0}_{i}$ and $\rho_{i}^{1}$ respectively in (8.13). This implies by (8.13) that $Q(t)=0$ for $t>0$ for sufficiently small. Hence the uniqueness follows by iterating this method. ∎

9. Appendix

Lemma A**.**

The following inequalities hold true:

• (a)

(Biler-Hebisch-Nadzieja type inequality [BCK*+*15]) For any $\epsilon>0$ there exists $L_{\epsilon}>0$ such that

[TABLE]

for all $\rho\in L^{1}_{+}(\mathbb{R}^{2})$ such that $\rho\ln\rho\in L^{1}(\mathbb{R}^{2})$ and $\frac{\nabla\rho}{\rho}\in L^{2}(\mathbb{R}^{2},\rho;\mathbb{R}^{2})$

• (b)

(Carleman Estimate [BDP06]) For any $\rho\in L^{1}_{+}(\mathbb{R}^{2})$ if $\int_{\mathbb{R}^{2}}\rho\ln\rho\ dx<+\infty$ and $M_{2}(\rho)<+\infty$ then

[TABLE]

9.1. Compactness Lemmas

We will also use the following compactness result whose proof can be found in [AGS05, Theorem $5.4.4$]:

Proposition A** (Compactness of vector fields).**

Let $\Omega$ be an open set in $\mathbb{R}^{N}$. If $\{\mu_{m}\}_{m}$ is a sequence of probability measures in $\Omega$ narrowly converging to $\mu$ (in duality with $C_{b}(\mathbb{R}^{N}),$ continuous bounded functions) and $\{v_{m}\}_{m}$ is a sequence of vector fields in $L^{2}(\Omega,\mu_{m};\mathbb{R}^{N})$ satisfying

[TABLE]

then there exists a vector field $v\in L^{2}(\Omega,\mu;\mathbb{R}^{N})$ such that

[TABLE]

and satisfy

[TABLE]

We recall a refined Arzelà-Ascoli’s compactness theorem obtained in [AGS05, Proposition $3.3.1$]: Let $(S,d)$ be a complete metric space and $\sigma$ be an Hausdorff topology on $S$ compatible with $d,$ in the sense that $\sigma$ is weaker than the topology induced by $d$ and $d$ is sequentially $\sigma$-lower semicontinuous:

[TABLE]

Theorem A** (Refined Arzelà-Ascoli).**

Let $T>0,$ let $K\subset\mathcal{S}$ be a sequentially compact set with respect to the topology $\sigma$ and let $u_{m}:[0,T]\rightarrow\mathcal{S}$ be curves such that

[TABLE]

for a symmetric function $w:[0,T]\times[0,T]\rightarrow[0,\infty),$ such that

[TABLE]

where $\mathcal{B}$ is an (at most) countable subset of $[0,T].$ Then there exists an increasing subsequence $p=m(p)$ and a limit curve $u:[0,T]\rightarrow\mathcal{S}$ such that

[TABLE]

The ensuing compactness lemma in $L^{2}(0,T;X)$ is a particular case of the compactness results obtained by J. Simon [Sim87, Lemma $9$]:

Lemma B** (Compactness in $L^{2}(0,T;X)$).**

Let $X\subset B\subset Y$ be Banach spaces such that $X\subset B$ is compact. If a family $F$ is bounded in $L^{2}(0,T;X)$ and relatively compact in $L^{2}(0,T;Y)$ then $F$ is relatively compact in $L^{2}(0,T,B).$