\newconstantfamily

epssymbol=ε

Long-time Asymptotic Behavior of Nonlinear Fokker-Planck Type Equations with Periodic Boundary Conditions

Yekaterina Epshteyn Department of Mathematics, The University of Utah, Salt Lake City, UT 84112, USA epshteyn@math.utah.edu , Chun Liu Department of Applied Mathematics, Illinois Institute of Technology. Chicago, IL 60616, USA cliu124@iit.edu and Masashi Mizuno Department of Mathematics, College of Science and Technology, Nihon University, Tokyo 101-8308 JAPAN mizuno.masashi@nihon-u.ac.jp

Abstract.

In this paper, we study the asymptotic behavior of a class of nonlinear Fokker-Planck type equations in a bounded domain with periodic boundary conditions. The system is motivated by our study of grain boundary dynamics, especially under the non-isothermal environments. To obtain the long time behavior of the solutions, in particular, the exponential decay, the kinematic structures of the systems are investigated using novel reinterpretation of the classical entropy method.

Key words and phrases:

Nonlinear Fokker-Planck equations, entropy methods, general diffusion.

1. Introduction

In this paper, we will discuss a specific class of Fokker-Planck systems that can be viewed as a special type of generalized diffusion models in the framework of the energetic-variational approach [24, 21, 17]. Such systems are determined by the kinematic transport of the probability density function, the free energy functional and the dissipation (entropy production), [2, 37]. In particular we are interested in the nonlinear equations with non-homogeneous diffusion and mobility in a finite bounded domain with periodic boundary conditions.

These Fokker-Planck type models are motivated by the studies of the macroscopic behavior of the systems that involve various fluctuations [35, 23, 15, 13, 11, 14, 27]. As in our previous work, we systematically studied such Fokker-Planck type systems as a part of grain growth models in polycrystalline materials, e.g. [4, 7, 3, 20].

One of the important goals of such study is to develop accurate mathematical models that take into account critical events, such as the grain disappearance or nucleation, grain boundary disappearance, facet interchange, and splitting of unstable junctions during coarsening process of microstructure. For example, the recent model derived in [20] under assumption of isothermal thermodynamics can be viewed as a further extension of a simplified 1D critical event model studied in [5, 4, 7, 3]. In [20], we have established the long time asymptotic results of the corresponding Fokker-Planck solutions, in terms of the joint probability density function of misorientations (a difference in the orientation between two neighboring grains that share a grain boundary) and triple junctions (triple junctions are where three grain boundaries meet), as well as the relation to the marginal probability density of misorientations. For an equilibrium configuration of a boundary network, we obtained the explicit local algebraic relationships, the generalized Herring Condition formula, as well as a novel relationship that connects grain boundary energy density with the geometry of the grain boundaries that share a triple junction.

The nonlinear Fokker-Planck equations proposed in this work also appear as a part of our current study of important case of non-isothermal thermodynamics [33, 9, 38]. Such Fokker-Planck systems are derived with applications to macroscopic models for grain boundary dynamics in polycrystalline materials [19, 18, 6, 17].

Most existing mathematical analysis work of the Fokker-Planck models is developed for the simplified linear cases only. This is especially true for the well-known entropy methods developed for the asymptotic analysis of such equations, e.g. [1, 28, 32, 12]. The classical entropy methods rely on the specific algebraic structures of the system, under convexity assumptions of the potential functions and consider systems in unbounded domains.

In this paper, we study the generalized nonlinear Fokker-Planck models, with inhomogeneous diffusion and/or mobility parameters in a bounded domain with periodic boundary conditions. This geometric constraint is relevant to our underlying grain boundary applications, together with the non-convexity constraints of the potential functions. Here we want to point out that in the paper [16], we had overlooked this crucial assumption in our mathematical analysis. While the results there are valid, the mathematical analysis becomes much more involved with less restrictive assumptions on the models. This is demonstrated in our current paper. Moreover, the mathematical analysis results in this paper are stronger than in [16] and in close agreement with the numerical results in [16] in the absence of the convexity conditions.

The paper is organized as follows. Below, we formulate the nonlinear Fokker-Planck model with the inhomogeneous diffusion and variable mobility parameters, introduce notations and some basic lemmas needed for the later sections. In Section 2, we first illustrate large time asymptotic analysis for the homogeneous case. In this case, the equations become the usual linear Fokker-Planck model. We employ the idea of the entropy method in terms of the velocity field of the solution. In Sections 3-4, we extend the analysis to the Fokker-Planck model with the inhomogeneous diffusion without or with variable mobility parameters respectively. Some conclusion remarks are given in Section 5.

Let us start with the following Fokker-Planck type of equation subject to the periodic boundary condition on a domain $\Omega=[0,1)^{n}\subset\mathbb{R}^{n}$

(1)

\left\{\begin{aligned} &\frac{\partial f}{\partial t}-\operatorname{div}\left(\frac{f}{\pi(x,t)}\nabla\left(D(x)\log f+\phi(x)\right)\right)=0,&\quad&x\in\Omega,t>0,\\ &f(x,0)=f_{0}(x),&\quad&x\in\Omega.\end{aligned}\right.

Here $D=D(x):\Omega\rightarrow\mathbb{R}$ , $\pi=\pi(x,t):\Omega\times[0,\infty)\rightarrow\mathbb{R}$ are given positive periodic functions with respect to $\Omega$ and $\phi=\phi(x):\Omega\rightarrow\mathbb{R}$ is a given periodic function with respect to $\Omega$ . The periodic boundary condition for $f$ means,

(2)

\nabla^{l}f(x_{b,1},t)=\nabla^{l}f(x_{b,2},t),

for

(3)

\begin{split}x_{b,1}&=(x_{1},x_{2},\ldots,x_{k-1},1,x_{k+1},\ldots,x_{n}),\\ x_{b,2}&=(x_{1},x_{2},\ldots,x_{k-1},0,x_{k+1},\ldots,x_{n})\in\partial\Omega,\end{split}

$t>0$ and $l=0,1,2,\ldots$ . In other words, $f$ can be smoothly extended to a function on the entire space $\mathbb{R}^{n}$ with the condition $f(x,t)=f(x+e_{j},t)$ for $x\in\mathbb{R}^{n}$ , $t>0$ and $j=1,2,\ldots,n$ , where $e_{j}=(0,\ldots,1,\ldots,0)$ , with the 1 in the $j$ th place. Note that, the periodic boundary condition for the function $f(x,t)$ is equivalent to the condition that $f(x,t)$ is the function on the $n$ -dimensional torus for $t>0$ . The periodic function is defined in the same way. The meaning of the periodic boundary condition for the Fokker-Planck equation can be seen in [34, §4.1, p.90].

The above equation can be viewed as a generalized diffusion in the general framework of energetic variational approaches [17, 24, 31, 39, 40]. One can see this by introducing a virtual velocity field $\bm{u}$ ,

(4)

\bm{u}=-\frac{1}{\pi(x,t)}\nabla\left(D(x)\log f+\phi(x)\right)

and rewrite the (1) in a equivalent form involving the following kinematic continuity equation (conservation of mass):

(5)

\left\{\begin{aligned} &\frac{\partial f}{\partial t}+\operatorname{div}(f\bm{u})=0,&\quad&x\in\Omega,\quad t>0,\\ &f(x,0)=f_{0}(x),&\quad&x\in\Omega.\end{aligned}\right.

The form of the first equation in (5) will motivate us to extend various conventional methods established for various fluids and diffusions to the nonlinear Fokker-Planck model with inhomogeneous temperature parameter $D(x)$ .

Next, from (5) together with integration by parts and with the periodic boundary condition, it is easy to obtain that,

(6)

\frac{d}{dt}\int_{\Omega}f\,dx=\int_{\Omega}\frac{\partial f}{\partial t}\,dx=-\int_{\Omega}\operatorname{div}(f\bm{u})\,dx=0.

Therefore, if $f_{0}$ is a probability density function on $\Omega$ , we have,

(7)

\int_{\Omega}f\,dx=\int_{\Omega}f_{0}\,dx=1.

Let $F$ be a free energy defined by,

(8)

F[f]:=\int_{\Omega}\left(D(x)f(\log f-1)+f\phi(x)\right)\,dx.

Hence, we can establish the energy law for (5). Hereafter, $|\cdot|$ denotes the standard Euclidean vector norm.

Proposition 1.1.

Let $f$ be a solution of the periodic boundary value problem (5), $\bm{u}$ be the velocity vector defined in (4), and let $F$ be a free energy defined in (8). Then, for $t>0$ ,

(9)

\frac{d}{dt}F[f](t)=-\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx=:-D_{\mathrm{Dis}}[f](t).

Proof.

Note that $D(x)$ and $\phi(x)$ are independent of $t$ , so direct computation yields

(10)

\frac{d}{dt}F[f](t)=\int_{\Omega}(D(x)\log f+\phi(x))\frac{\partial f}{\partial t}\,dx.

Using (1) together with periodic boundary condition and integration by parts, we obtain

(11)

\begin{split}\int_{\Omega}(D(x)\log f+\phi(x))\frac{\partial f}{\partial t}\,dx&=-\int_{\Omega}(D(x)\log f+\phi(x))\operatorname{div}(f\bm{u})\,dx\\ &=\int_{\Omega}\nabla(D(x)\log f+\phi(x))\cdot(f\bm{u})\,dx.\end{split}

From (4), $\nabla(D(x)\log f+\phi(x))=-\pi(x,t)\bm{u}$ hence

(12)

\int_{\Omega}\nabla(D(x)\log f+\phi(x))\cdot(f\bm{u})\,dx=-\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx.

Thus, (9) is proved by combining (10), (11), and (12). ∎

Throughout this paper, we assume the Hölder regularity with $0<\alpha<1$ for coefficients $\pi(x,t)$ , $D(x)$ , $\phi(x)$ and initial datum $f_{0}$ ,

(13)

\begin{split}&\pi\in C_{\mathrm{per}}^{1+\alpha,(1+\alpha)/2}(\Omega\times[0,T)),\ D\in C_{\mathrm{per}}^{2+\alpha}(\Omega),\\ &\phi\in C_{\mathrm{per}}^{2+\alpha}(\Omega),\ f_{0}\in C_{\mathrm{per}}^{2+\alpha}(\Omega),\end{split}

where

C_{\mathrm{per}}^{2+\alpha}(\Omega):=\{g\in C^{2+\alpha}(\Omega):g\ \text{is a periodic function on }\Omega\},

C_{\mathrm{per}}^{1+\alpha,(1+\alpha)/2}(\Omega\times[0,T))\\ :=\{g\in C^{1+\alpha,(1+\alpha)/2}(\Omega\times[0,T)):\\ g(\cdot,t)\ \text{is a periodic function on }\Omega\ \text{for}\ t>0\}.

In this paper, we will consider the classical solutions of (1) defined below. In principle, they will be smooth enough so that all the derivatives and integrations evolved in the equations and the estimates will make sense (see [17, 22, 29, 30]).

Definition 1.2.

A periodic function in space $f=f(x,t)$ is a classical solution of the problem (1) in $\Omega\times[0,T)$ , subject to the periodic boundary condition, if $f\in C^{2,1}(\Omega\times(0,T))\cap C^{1,0}(\overline{\Omega}\times[0,T))$ , $f(x,t)>0$ for $(x,t)\in\Omega\times[0,T)$ , and satisfies equation (1) in a classical sense.

Remark 1.3.

In [17], we discussed the local well-posedness of the system with the no-flux boundary condition in the Hölder space settings. We also mentioned the local well-posedness in [16, Proposition 1.5]. Recently, global well-posedness of the problem (1) was studied in [8]. Above, we recalled the definition of the corresponding Hölder spaces but we note that the arguments in this paper are not necessarily in the Hölder space settings.

Next we will prescribe those constants in terms of $C^{\prime}s$ . associated with various quantities. This is important for the computations and estimates in the later sections.

(1)

The bounds for the initial data:

(14) $0<\Cl{const:InitMin}\leq f_{0}(x,t)\leq\Cl{const:InitMax}.$
(2)

The lower bound for the diffusion $D(x)$ , $\Cr{const:D_{M}in}\geq 1$ such that

(15) $D(x)\geq\Cl{const:D_{M}in}.$
(3)

The positive bounds for the mobility:

(16) $0<\Cl{const:Pi_{M}in}\leq\pi(x,t)\leq\Cl{const:Pi_{M}ax}.$
(4)

The bound for the derivatives of the mobility:

(17) $|\pi_{t}(x,t)|\leq\Cl{const:Pi_{T}ime},\ \text{and}$

(18) $|\nabla\pi(x,t)|\leq\Cl{const:grad_{P}i}.$
(5)

The bound of the derivative of $D(x)$ :

(19) $|\nabla D(x)|\leq\Cl{const:grad_{D}}.$
(6)

The lower bound of the Hessian of $\phi$ :

(20) $\nabla^{2}\phi(x)\geq-\lambda I,\quad\text{for}\quad x\in\Omega.$

It is clear that these constants/parameters will also satisfy various relations among themselves and they are also associated with the original system.

Lemma 1.4.

For a positive function $D(x)$ , the following is true:

(21)

|D(x)|\leq\Cr{const:D_{M}in}+\sqrt{n}\Cr{const:grad_{D}},

for $\Cr{const:D_{M}in}$ and $\Cr{const:grad_{D}}$ defined earlier and $\Omega$ being a unit square/cube.

Proof.

For any $x,y\in\Omega$ , we first note that

(22)

D(x)=D(y)+\int_{0}^{1}\frac{d}{d\varepsilon}D(\varepsilon x+(1-\varepsilon)y)\,d\varepsilon.

From (19) and $\Omega$ being a unit square/cube, we have

(23)

\left|\frac{d}{d\varepsilon}D(\varepsilon x+(1-\varepsilon)y)\right|=\left|\nabla D(\varepsilon x+(1-\varepsilon)y)\cdot(x-y)\right|\leq\sqrt{n}\Cr{const:grad_{D}}.

By the triangle inequality and (22), we get

(24)

|D(x)|\leq|D(y)|+\sqrt{n}\Cr{const:grad_{D}}.

Therefore, (21) is deduced by taking infimum with respect to $y$ in (24). ∎

We look at an equilibrium solution of (1) in the form

(25)

f^{\mathrm{eq}}(x):=\exp\left(-\frac{\phi(x)-\Cl{const:Feq}}{D(x)}\right),

where the constant $\Cr{const:Feq}$ is determined as

(26)

\int_{\Omega}f^{\mathrm{eq}}(x)\,dx=1.

Here one can derive the relation between the constant $\Cr{const:Feq}$ and the potential function $\phi$ .

Lemma 1.5.

Let $\Cr{const:Feq}$ be in (25). Then

(27)

|\Cr{const:Feq}|\leq\|\phi\|_{L^{\infty}(\Omega)}.

Proof.

By the mean value theorem and (26), there exists $x_{0}\in\Omega$ such that

(28)

f^{\mathrm{eq}}(x_{0})=\frac{1}{|\Omega|}\int_{\Omega}f^{\mathrm{eq}}(x)\,dx=1.

Thus,

(29)

\exp\left(-\frac{\phi(x_{0})-\Cr{const:Feq}}{D(x_{0})}\right)=1

hence $\phi(x_{0})=\Cr{const:Feq}$ . The estimate (27) is easily deduced. ∎

From (27), it follows immediately that

Lemma 1.6.

Let $f^{\mathrm{eq}}$ be defined by (25). Then, for $x\in\Omega$ ,

(30)

\exp\left(-\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}\right)\leq f^{\mathrm{eq}}(x)\leq\exp\left(\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}\right).

In [16, Proposition 1.6], we have derived the following maximum principle which is crucial to our analysis.

Proposition 1.7.

Let the coefficients $\pi(x,t)$ , $\phi(x)$ , $D(x)$ , and a positive probability density function $f_{0}(x)$ satisfy the strong positivity (15), (16), (14) and the Hölder regularity (13) for $0<\alpha<1$ . Let $f$ be a classical solution of (1), then the following holds:

(31)

\begin{split}&\exp\left(\frac{1}{D(x)}\min_{y\in\Omega}\left(D(y)\log\frac{f_{0}(y)}{f^{\mathrm{eq}}(y)}\right)\right)f^{\mathrm{eq}}(x)\\ &\quad\leq f(x,t)\leq\exp\left(\frac{1}{D(x)}\max_{y\in\Omega}\left(D(y)\log\frac{f_{0}(y)}{f^{\mathrm{eq}}(y)}\right)\right)f^{\mathrm{eq}}(x),\end{split}

for $x\in\Omega$ , $t>0$ .

From Proposition 1.7, we have the following logarithm estimates. In particular, we will show that the bound of the solution is independent of the diffusion bound $\Cr{const:D_{M}in}$ .

Corollary 1.8.

As in the same assumptions in Proposition 1.7, there exists a positive constant $\Cl{const:log_{f}}$ which depends only on $n$ , the initial datum $f_{0}$ (in terms of $\Cr{const:InitMin},\ \Cr{const:InitMax}$ ), the gradient of $D$ (in terms of $\Cr{const:grad_{D}}$ ), and the bounds of the potential $\|\phi\|_{L^{\infty}(\Omega)}$ such that

(32)

|\log f(x,t)|\leq\Cr{const:log_{f}}.

Moreover, $\Cr{const:log_{f}}$ is independent to the diffusion bound $\Cr{const:D_{M}in}$ .

Proof.

By (31), we have by (15) that,

(33)

\begin{split}&\frac{1}{D(x)}\min_{y\in\Omega}\left(D(y)\log\frac{f_{0}(y)}{f^{\mathrm{eq}}(y)}\right)+\log f^{\mathrm{eq}}(x)\\ &\quad\leq\log f(x,t)\leq\left(\frac{1}{D(x)}\max_{y\in\Omega}\left(D(y)\log\frac{f_{0}(y)}{f^{\mathrm{eq}}(y)}\right)\right)+\log f^{\mathrm{eq}}(x).\end{split}

From here we have the following by using the assumption (15):

(34)

\begin{split}|\log f(x,t)|&\leq\frac{1}{\Cr{const:D_{M}in}}\|D\|_{L^{\infty}(\Omega)}(\|\log f_{0}\|_{L^{\infty}(\Omega)}\\ &\quad+\|\log f^{\mathrm{eq}}\|_{L^{\infty}(\Omega)})+\|\log f^{\mathrm{eq}}\|_{L^{\infty}(\Omega)}.\end{split}

Using (21) and (30), we obtain,

(35)

\begin{split}&\frac{1}{\Cr{const:D_{M}in}}\|D\|_{L^{\infty}(\Omega)}(\|\log f_{0}\|_{L^{\infty}(\Omega)}+\|\log f^{\mathrm{eq}}\|_{L^{\infty}(\Omega)})+\|\log f^{\mathrm{eq}}\|_{L^{\infty}(\Omega)}\\ &\leq\frac{1}{\Cr{const:D_{M}in}}(\Cr{const:D_{M}in}+\sqrt{n}\Cr{const:grad_{D}})\left(\|\log f_{0}\|_{L^{\infty}(\Omega)}+\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}\right)+\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}\\ &=\left(1+\frac{\sqrt{n}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\right)\left(\|\log f_{0}\|_{L^{\infty}(\Omega)}+\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}\right)+\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}.\end{split}

Since $\Cr{const:D_{M}in}\geq 1$ , we have

(36)

\begin{split}&\left(1+\frac{\sqrt{n}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\right)\left(\|\log f_{0}\|_{L^{\infty}(\Omega)}+\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}\right)+\frac{2\|\phi\|_{L^{\infty}(\Omega)}}{\Cr{const:D_{M}in}}\\ &\leq(1+\sqrt{n}\Cr{const:grad_{D}})(\|\log f_{0}\|_{L^{\infty}(\Omega)}+2\|\phi\|_{L^{\infty}(\Omega)})+2\|\phi\|_{L^{\infty}(\Omega)}=:\Cr{const:log_{f}}.\end{split}

Here, the constant $\Cr{const:log_{f}}>0$ depends only on $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ , $\Cr{const:grad_{D}}$ and $\|\phi\|_{L^{\infty}(\Omega)}$ . ∎

Remark 1.9.

We emphasize that from the above corollary, we can see that $\Cr{const:log_{f}}$ can be taken uniform with respect to $\Cr{const:D_{M}in}$ , while the solution $f$ of the PDE (1) certainly depends on the diffusion $D(x)$ . We also note that (32) is equivalent to

(37)

e^{-\Cr{const:log_{f}}}\leq f(x,t)\leq e^{\Cr{const:log_{f}}}

for all $x\in\Omega$ and $t>0$ , namely uniform lower and upper bound of $f$ with respect to $\Cr{const:D_{M}in}$ . This immediately yields the following Harnack type estimate:

(38)

\frac{\max_{x\in\Omega,t>0}f(x,t)}{\min_{x\in\Omega,t>0}f(x,t)}\leq e^{2\Cr{const:log_{f}}}.

In addition, we would need to use later the estimate (42) of $|\operatorname{div}\bm{u}|$ in terms of $|\nabla\bm{u}|$ below, where

(39)

|\nabla\bm{u}|^{2}=\sum_{k,l=1}^{n}\left(\frac{\partial u^{k}}{\partial x_{l}}\right)^{2},\qquad\bm{u}=(u^{1},u^{2},\ldots,u^{n}).

To obtain (42), we use the following Jensen’s inequality [26, V.19], [36, Theorem 4.3].

Proposition 1.10 ([26, V.19]).

Let f be a function from $\mathbb{R}$ to $(-\infty,\infty]$ . Then $f$ is convex if and only if,

(40)

f(\lambda_{1}x_{1}+\cdots+\lambda_{m}x_{m})\leq\lambda_{1}f(x_{1})+\cdots+\lambda_{m}f(x_{m})

whenever $\lambda_{1},\ldots,\lambda_{m}\geq 0$ , $\lambda_{1}+\cdots+\lambda_{m}=1$ , and $x_{1},\ldots,x_{m}\in\mathbb{R}$ .

Applying Jensen’s inequality (40) to convex function $x^{2}$ , we have that,

(41)

\left(\frac{a_{1}+\cdots+a_{n}}{n}\right)^{2}\leq\frac{1}{n}\left(a_{1}^{2}+\cdots+a_{n}^{2}\right)

for $a_{1},\ldots,a_{n}\in\mathbb{R}$ . Using, $a_{j}=\frac{\partial u^{j}}{\partial x_{j}}$ in (41), we obtain,

(42)

|\operatorname{div}\bm{u}|^{2}\leq n\sum_{j=1}^{n}\left(\frac{\partial u^{j}}{\partial x_{j}}\right)^{2}\leq n|\nabla\bm{u}|^{2}.

2. Homogeneous diffusion case

In this section, we consider the cases with both diffusion and mobility being homogeneous. In this case, $D$ is a constant, and without loss of generality, we choose $\pi\equiv 1$ and take $\Omega=[0,1)^{n}\subset\mathbb{R}^{n}$ .

Here, we study the following evolution equation with periodic boundary conditions.

(43)

\left\{\begin{aligned} &\frac{\partial f}{\partial t}+\operatorname{div}\left(f\bm{u}\right)=0,&\quad&x\in\Omega,\quad t>0,\\ &\bm{u}=-\nabla\left(D\log f+\phi(x)\right),&\quad&x\in\Omega,\quad t>0,\\ &f(x,0)=f_{0}(x),&\quad&x\in\Omega.\end{aligned}\right.

The free energy $F$ associated with (43) will take the form :

(44)

F[f](t)=\int_{\Omega}(Df(x,t)(\log f(x,t)-1)+f(x,t)\phi(x))\,dx.

As we had presented in Proposition 1.1, the following energy law will hold for any solution $f$ of (43):

(45)

\frac{d}{dt}F[f](t)=-\int_{\Omega}|\bm{u}|^{2}f\,dx=-D_{\mathrm{dis}}[f](t).

In this setting, equation (43) is a linear parabolic equation. One can use various established methods and techniques to investigate the long-time asymptotic behavior for a solution of (43). For instance, making the change of variable

f(x,t)=g(x,t)\exp\left(-\frac{\phi(x)}{D}\right),

one may associate the original equation with a self-adjoint operator $Lg=D\Delta g-\nabla\phi\cdot\nabla g$ on $L^{2}(\Omega,e^{-\frac{\phi}{D}}\,dx)$ (See [20]).

In this paper, we plan to study the long time behavior of the solutions, especially the exponential decay through the investigation of higher order time derivative of the free energy functional. We want to point out that the current method is related to the entropy method that had been developed previously for various Fokker-Planck type of equations in unbounded domains. We consider here bounded domain and our approach takes the full advantage of the kinematic structures, such as looking at the velocity variable $\bm{u}$ .

Theorem 2.1.

Given the potential $\phi$ and the domain $\Omega$ in $\mathbb{R}^{n}$ in (43), for any $\gamma>0$ , there exists $\Cr{const:D_{M}in}\geq 1$ which depends on $n$ , the potential $\phi$ (in terms of the lower bound of the Hessian $\lambda$ defined in (20) and $\|\phi\|_{L^{\infty}(\Omega)}$ ), the bounds of the initial data $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined on (14), and $\gamma$ , such that if (15) holds and,

(46)

\int_{\Omega}|\nabla(D\log f_{0}+\phi(x))|^{2}f_{0}\,dx=:\Cr{const:2.InitialEnergy}<\infty,

then, the following dissipation rate decays exponentially in time:

(47)

\int_{\Omega}|\bm{u}|^{2}f\,dx\leq\Cl{const:2.InitialEnergy}e^{-\gamma t}.

Remark 2.2.

The above theorem states that for given $\phi$ and the domain $\Omega$ , one can obtain any decay rate $\gamma$ by choosing the diffusion constant $D$ large enough. Conversely, from the proof of the theorem, we can show that for large enough $D$ (estimate is given in (58) later), the dissipation rate of the system will decay exponentially in time.

Remark 2.3.

In the conventional entropy method, the function $\phi$ is usually assumed to be convex, and the domain is the whole space. In this paper, with a bounded domain subject to the periodic boundary conditions, we can assume the Hessian of $\phi$ to be bounded below. In particular, we can treat cases with non-convex function $\phi$ .

To prove Theorem 2.1, we first recall the second derivative of the free energy [16].

Proposition 2.4 ([16, Proposition 2.9]).

For the free energy $F$ defined in (44), and $f$ be a solution of (43), we obtain:

(48)

\frac{d^{2}}{dt^{2}}F[f](t)=2\int_{\Omega}(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})f\,dx+2\int_{\Omega}D|\nabla\bm{u}|^{2}f\,dx.

Since $\nabla^{2}\phi$ might be non-positive, our plan is to use the second term, $\int_{\Omega}D|\nabla\bm{u}|^{2}f\,dx$ to control the right hand side of (48). We first establish the following inequality.

Lemma 2.5.

Let $f$ be a solution of (43), and let $\bm{u}$ be given as in (43). There exists $\Cl{const:2.Sobolev}>0$ which depends only on $n$ , $f_{0}$ (in terms of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14)), and $\|\phi\|_{L^{\infty}(\Omega)}$ such that

(49)

\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\leq\Cr{const:2.Sobolev}\left(\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\right)^{\frac{1}{2}},

where $2<p^{*}<\infty$ for $n=1,2$ , and $\frac{1}{p^{*}}=\frac{1}{2}-\frac{1}{n}$ for $n\geq 3$ .

Proof.

By the Sobolev inequality ([25, Lemma 7.12, 7.16],[10, p.313]), there is $\Cl{const:Sobolev}>0$ which depends only on $n$ such that,

\left(\int_{\Omega}|\bm{v}-\overline{\bm{v}}|^{p^{*}}\,dx\right)^{\frac{1}{p^{*}}}\leq\Cr{const:Sobolev}\left(\int_{\Omega}|\nabla\bm{v}|^{2}\,dx\right)^{\frac{1}{2}}

for $\bm{v}\in W^{1,2}(\Omega)$ , where $\overline{\bm{v}}$ is the integral mean of $\bm{v}$ , namely

\overline{\bm{v}}=\frac{1}{|\Omega|}\int_{\Omega}\bm{v}\,dx.

Here, $2<p^{*}<\infty$ for $n=1,2$ , and $\frac{1}{p^{*}}=\frac{1}{2}-\frac{1}{n}$ for $n\geq 3$ .

Since $\bm{u}$ has the scalar potential, we can compute,

(50)

\begin{split}\int_{\Omega}\bm{u}\,dx&=-\int_{\Omega}\nabla\left(D\log f+\phi(x)\right)\,dx\\ &=-\int_{\partial\Omega}\left(D\log f+\phi(x)\right)\nu\,d\sigma=\bm{0}.\end{split}

Hence we obtain by taking $\bm{v}=\bm{u}$ that,

\begin{split}\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}&\leq\left(\max_{(x,t)\in\Omega\times[0,\infty)}f\right)^{\frac{1}{p^{*}}}\left(\int_{\Omega}|\bm{u}|^{p^{*}}\,dx\right)^{\frac{1}{p^{*}}}\\ &\leq\Cr{const:Sobolev}\left(\max_{(x,t)\in\Omega\times[0,\infty)}f\right)^{\frac{1}{p^{*}}}\left(\int_{\Omega}|\nabla\bm{u}|^{2}\,dx\right)^{\frac{1}{2}}\\ &\leq\Cr{const:Sobolev}\frac{(\max_{(x,t)\in\Omega\times[0,\infty)}f)^{\frac{1}{p^{*}}}}{(\min_{(x,t)\in\Omega\times[0,\infty)}f)^{\frac{1}{2}}}\left(\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\right)^{\frac{1}{2}}.\end{split}

Note from (32) (see Remark 1.9) that $e^{-\Cr{const:log_{f}}}\leq f(x,t)\leq e^{\Cr{const:log_{f}}}$ for all $x\in\Omega$ and $t>0$ . This inequality yields

\frac{(\max_{(x,t)\in\Omega\times[0,\infty)}f)^{\frac{1}{p^{*}}}}{(\min_{(x,t)\in\Omega\times[0,\infty)}f)^{\frac{1}{2}}}\leq e^{\Cr{const:log_{f}}\left(\frac{1}{p^{*}}+\frac{1}{2}\right)}.

Letting $\Cr{const:2.Sobolev}=\Cr{const:Sobolev}e^{\Cr{const:log_{f}}\left(\frac{1}{p^{*}}+\frac{1}{2}\right)}$ , we obtain (49). ∎

Remark 2.6.

In this section, $D$ is constant, which means $\nabla D=0$ hence $\Cr{const:log_{f}}$ depends on the $L^{\infty}$ norms of the initial data $f_{0}$ in term of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ and potential $\|\phi\|_{L^{\infty}(\Omega)}$ , but is independent of $\Cr{const:grad_{D}}$ defined in (19).

Remark 2.7.

In this proof, it is important that $\bm{u}$ is a potential gradient. For (1), when the mobility $\pi(x,t)$ is not constant, the velocity $\bm{u}$ is not a potential gradient anymore, and we will have to derive more estimates as we will do in Section 4.

From here, we can show the following Poincare type inequality for the velocity $\bm{u}$ .

Lemma 2.8.

Let $f$ be a solution of (43), and let $\bm{u}$ be given as in (43). There exists a constant $\Cl{const:2.Poincare}>0$ which depends only on $n$ , the $L^{\infty}$ norms of the initial data $f_{0}$ (in terms of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined on (14)), and $\|\phi\|_{L^{\infty}(\Omega)}$ such that,

(51)

\int_{\Omega}|\bm{u}|^{2}f\,dx\leq\Cr{const:2.Poincare}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx.

Proof.

We show (51) for the case $n\geq 3$ . First, we use the Hölder inequality and (7) that

(52)

\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{\frac{1}{2}}\leq\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\left(\int_{\Omega}f\,dx\right)^{\frac{1}{2}-\frac{1}{p^{*}}}=\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}

where we select $p^{*}>2$ to satisfy $\frac{1}{p^{*}}=\frac{1}{2}-\frac{1}{n}$ so we can employ the Sobolev’s inequality (49). Next, we use (49) to have that,

(53)

\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\leq\Cr{const:2.Sobolev}\left(\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\right)^{\frac{1}{2}}.

Letting $\Cr{const:2.Poincare}=\Cr{const:2.Sobolev}^{2}$ , we obtain (51). Note that $\Cr{const:log_{f}}$ depends on $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ and $\|\phi\|_{L^{\infty}(\Omega)}$ so we can take $\Cr{const:2.Poincare}$ which depends only on $n$ , $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ and $\|\phi\|_{L^{\infty}(\Omega)}$ .

For the cases $n=1,2$ , (49) holds for any $2<p^{*}<\infty$ . If, for instance, by taking $p^{*}=6$ , we will be able to use the exact same estimates as in the case $n=3$ above to get the result. ∎

Now, we are in a position to derive the time derivative of the integral $|\bm{u}|^{2}$ .

Proposition 2.9.

Let $f$ be a solution of (43), and let $\bm{u}$ be given as in (43). For any $\gamma>0$ , there exists $\Cr{const:D_{M}in}\geq 1$ which depends only on $n$ , $\lambda$ , $\gamma$ , $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined on (14), and $\|\phi\|_{L^{\infty}(\Omega)}$ such that

(54)

\frac{d}{dt}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)\leq-\gamma\int_{\Omega}|\bm{u}|^{2}f\,dx.

Proof.

From the second derivative of the free energy (48) and the lower bounds of $\nabla^{2}\phi$ (20), we have

(55)

\frac{d^{2}}{dt^{2}}F[f](t)\geq-2\lambda\int_{\Omega}|\bm{u}|^{2}f\,dx+2D\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx.

Next, we use the Poincare type inequality (51) that

(56)

\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\geq\frac{1}{\Cr{const:2.Poincare}}\int_{\Omega}|\bm{u}|^{2}f\,dx.

Together with the two inequalities, we obtain

(57)

\frac{d^{2}}{dt^{2}}F[f](t)\geq\left(-2\lambda+\frac{2D}{\Cr{const:2.Poincare}}\right)\int_{\Omega}|\bm{u}|^{2}f\,dx.

Choose $\Cr{const:D_{M}in}\geq 1$ sufficiently large such that

(58)

-2\lambda+\frac{2\Cr{const:D_{M}in}}{\Cr{const:2.Poincare}}\geq\gamma.

Then, from the energy law (45), we have that,

(59)

-\frac{d}{dt}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)\geq\gamma\int_{\Omega}|\bm{u}|^{2}f\,dx.

This finishes the proof of the Proposation. ∎

Finally, we are in position to prove the main result of this Section, Theorem 2.1.

Proof of Theorem 2.1.

Applying the Gronwall inequality to the differential inequality (54), we obtain

(60)

\int_{\Omega}|\bm{u}|^{2}f\,dx\leq e^{-\gamma t}\int_{\Omega}|\nabla\left(D\log f_{0}+\phi(x)\right)|^{2}f_{0}\,dx.

Using (46), we obtain (47). ∎

In Section 2, in the case that $D$ and $\pi$ are constants, we derive the exponential decay of the time derivative of the free energy $F$ by construction of the differential inequality (54). The crucial idea is to use the Poincare type inequality (51) in Lemma 2.8. In the following sections, the Poincare type inequality will also play crucial roles in the study.

Remark 2.10.

We emphasize that the above argument is based on the kinematic structure of the equation (5) and the velocity $\bm{u}$ . This is in consistent with the conventional entropy methods in the whole domain cases, which work on the whole PDE itself. This approach is much straightforward and uses the physical relevant quantities.

Remark 2.11.

The result in Theorem 2.1 shows that for a given decay rate $\gamma>0$ , we may find the class of equations (43) with sufficiently large diffusion $D$ and bounded initial data $f_{0}$ (in terms of $D_{\mathrm{dis}}[f](0)<\infty$ ), then the system will evolve exponentially towards equilibrium. In particularly, we prove the exponential decay property of dissipation rate $D_{\mathrm{dis}}[f](t)$ , which appears in (45), with the given rate $\gamma$ . Conversely the proof of the theorem also give the sufficient condition in $D$ for the exponential decay for a system with bounded initial data. Moreover it is open question whether the result holds without assumption for the diffusion coefficient $D\geq\Cr{const:D_{M}in}$ to be large enough.

3. Inhomogeneous diffusion case

In this section, we consider the following evolution equation with inhomogeneous diffusion while the mobility remaining a constant.

(61)

\left\{\begin{aligned} &\frac{\partial f}{\partial t}+\operatorname{div}\left(f\bm{u}\right)=0,&\quad&x\in\Omega,\quad t>0,\\ &\bm{u}=-\nabla\left(D(x)\log f+\phi(x)\right),&\quad&x\in\Omega,\quad t>0,\\ &f(x,0)=f_{0}(x),&\quad&x\in\Omega,\end{aligned}\right.

with periodic boundary conditions. Without loss of generality, we take $\Omega=[0,1)^{n}\subset\mathbb{R}^{n}$ . We choose a strictly positive periodic function $D=D(x)$ with the positive lower bound, $\Cr{const:D_{M}in}\geq 1$ , i.e., $D(x)\geq\Cr{const:D_{M}in},$ for $x\in\Omega$ .

Again, the direct formal computations show that the free energy $F$ and the basic energy law (9) take the following specific forms,

(62)

F[f]:=\int_{\Omega}\left(D(x)f(\log f-1)+f\phi(x)\right)\,dx,

and

(63)

\frac{d}{dt}F[f](t)=-\int_{\Omega}|\bm{u}|^{2}f\,dx=:-D_{\mathrm{dis}}[f](t).

As discussed in [16], the energy law (63) with the free energy (62) carries all the physics of the system, and the system, together with the kinematic assumption of $f$ , will yield the equation (61) by the energetic variational approach. In particular, we note that inhomogeneity $D(x)$ is the source of the nonlinearity in (61). Like what we had proved in the previous section, Theorem 2.1, the following theorem states that for any function $\phi$ with bounded second derivatives, we can show exponential decay for the dissipation rate in the energy law provided the diffusion coefficient $D(x)$ is sufficiently large. However, besides assumptions made in Theorem 2.1, we will make additional assumptions on the gradients of $D(x)$ and $\phi$ , as well as assumption on the number of dimension $n$ .

Theorem 3.1.

Assume $n=1,2,3$ . For a fixed constant $\gamma>0$ , there exist positive constants $\Cr{const:D_{M}in}\geq 1$ and $\Cl{const:3.Initial_{E}nergy}>0$ which depend on dimension $n$ , the lower bound of Hessian of $\phi$ in terms of $\lambda$ defined in (20), $\gamma$ , the initial data $f_{0}$ (in terms of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined on (14)), the gradient of $D(x)$ (in terms of $\Cr{const:grad_{D}}$ defined in (19)), $\|\phi\|_{L^{\infty}(\Omega)}$ , and $\|\nabla\phi\|_{L^{\infty}(\Omega)}$ , such that if (15) holds and

(64)

\int_{\Omega}|\nabla(D(x)\log f_{0}+\phi(x))|^{2}f_{0}\,dx\leq\Cr{const:3.Initial_{E}nergy},

then, for $t>0$ , we have the following result:

(65)

\int_{\Omega}|\bm{u}|^{2}f\,dx\leq\Cr{const:3.Exponential_{C}oefficient}e^{-\gamma t},

for a positive constant $\Cl{const:3.Exponential_{C}oefficient}>0$ .

As in section 2, we have the following Sobolev-type inequality for the velocity $\bm{u}$ . Although the proof is the same as in Lemma 2.5, due to the fact that $\bm{u}$ is a potential gradient in this case, the constant $\Cr{const:3.Sobolev}$ , which is in Lemma 3.2, will depend on $\Cr{const:grad_{D}}$ defined in (19).

Lemma 3.2.

Let $f$ be a solution of (61), and let $\bm{u}$ be given as in (61). There exists $\Cr{const:3.Sobolev}>0$ which depends only on $n$ , $f_{0}$ (in terms of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14)), the gradient of $D(x)$ (in terms of $\Cr{const:grad_{D}}$ defined in (19)), and $\|\phi\|_{L^{\infty}(\Omega)}$ such that

(66)

\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\leq\Cl{const:3.Sobolev}\left(\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\right)^{\frac{1}{2}},

where $2<p^{*}<\infty$ for $n=1,2$ and $\frac{1}{p^{*}}=\frac{1}{2}-\frac{1}{n}$ for $n\geq 3$ .

With the extra dependence of the constants, the same proof in Lemma 2.8 will yield the following Poincare-type inequality for the velocity $\bm{u}$ .

Lemma 3.3.

Let $f$ be a solution of (61), and let $\bm{u}$ be given as in (61). There exists a constant $\Cl{const:3.Poincare}>0$ which depends only on $n$ , $f_{0}$ (in terms of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14)), the gradient of $D(x)$ (in terms of $\Cr{const:grad_{D}}$ defined in (19)), and $\|\phi\|_{L^{\infty}(\Omega)}$ such that

(67)

\int_{\Omega}|\bm{u}|^{2}f\,dx\leq\Cr{const:3.Poincare}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx.

Remark 3.4.

Compare to Lemma 2.8, the constants $\Cr{const:3.Sobolev}$ and $\Cr{const:3.Poincare}$ depend not only on $n$ , $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ , and $\|\phi\|_{L^{\infty}(\Omega)}$ , but also $\Cr{const:grad_{D}}$ , the upper bound of the gradient of $D(x)$ . We emphasize again that the constants above $\Cr{const:3.Sobolev}$ and $\Cr{const:3.Poincare}$ are uniform with respect to $\Cr{const:D_{M}in}$ .

In the next lemma, we obtain the following interpolation inequality.

Lemma 3.5.

Let $n=1,2,3$ . Let $f$ be a solution of (61), and let $\bm{u}$ be given as in (61). Then we have

(68)

\int_{\Omega}|\bm{u}|^{3}f\,dx\leq\frac{3}{4}\Cr{const:3.Sobolev}^{\frac{3}{2}}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx+\frac{1}{4}\Cr{const:3.Sobolev}^{\frac{3}{2}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.

Proof.

The proof follows from the same as those in [16, Lemma 3.14] (See also Lemma 4.7) together with Lemma 3.2. ∎

We next recall the second derivative of the free energy [16], which can be obtained by direct computation from the original system (61):

Proposition 3.6 ([16, Proposition 3.13]).

Let $f$ be a solution of (61) and let $\bm{u}$ be given as in (61). Then the following is true:

(69)

\begin{split}\frac{d^{2}}{dt^{2}}F[f](t)&=2\int_{\Omega}((\nabla^{2}\phi(x))\bm{u}\cdot\bm{u})f\,dx+2\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad-\int_{\Omega}(\log f-1)\nabla|\bm{u}|^{2}\cdot\nabla D(x)f\,dx\\ &\qquad-2\int_{\Omega}(1+\log f)\bm{u}\cdot\nabla D(x)\operatorname{div}\bm{u}f\,dx\\ &\qquad+2\int_{\Omega}\frac{1}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx\\ &\qquad+2\int_{\Omega}\frac{1}{D(x)}(\log f)^{2}\left(\bm{u}\cdot\nabla D(x)\right)^{2}f\,dx\\ &\qquad+2\int_{\Omega}\frac{1}{D(x)}\log f\left(\bm{u}\cdot\nabla D(x)\right)\left(\bm{u}\cdot\nabla\phi(x)\right)f\,dx.\end{split}

Below out of total 7 terms on the right-hand side of (69), we first consider the 3rd, 4th, and 7th integral of the right-hand side of (69). Notice the 2nd and 6th terms are with the right positive sign, while the 5th term includes the cubic order of $\bm{u}$ .

Lemma 3.7.

Let $f$ be a solution of (61), and let $\bm{u}$ be given as in (61). Then, we have

(70)

\begin{split}\frac{d^{2}}{dt^{2}}F[f](t)&\geq 2\int_{\Omega}\biggl{(}(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})\\ &\qquad\qquad-\Bigl{(}\frac{1}{2}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\Bigr{)}|\bm{u}|^{2}\biggr{)}f\,dx\\ &\quad+2\int_{\Omega}\biggl{(}1-\frac{(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}\biggr{)}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\quad+2\int_{\Omega}\frac{1}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx.\end{split}

Proof.

Note that the integrand of the 6th term of (69) is non-negative, we need to estimate the integrands of the 3rd, 4th, and 7th integrals of (69). We have by Cauchy’s inequality with a fixed $\varepsilon=1/4$ (See [22, p.662]) that,

(71)

\begin{split}|(\log f-1)\nabla|\bm{u}|^{2}\cdot\nabla D(x)f|&\leq 2(|\log f|+1)^{2}|\nabla\bm{u}|^{2}|\nabla D(x)|^{2}f+\frac{1}{2}|\bm{u}|^{2}f\\ &\leq\frac{2\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}D(x)|\nabla\bm{u}|^{2}f+\frac{1}{2}|\bm{u}|^{2}f,\end{split}

(72)

\begin{split}|2(1+\log f)\bm{u}\cdot\nabla D(x)\operatorname{div}\bm{u}f|&\leq 2n(|\log f|+1)^{2}|\nabla\bm{u}|^{2}|\nabla D(x)|^{2}f+\frac{1}{2}|\bm{u}|^{2}f\\ &\leq\frac{2n\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}D(x)|\nabla\bm{u}|^{2}f+\frac{1}{2}|\bm{u}|^{2}f,\end{split}

and

(73)

\left|\frac{2}{D(x)}\log f\left(\bm{u}\cdot\nabla D(x)\right)\left(\bm{u}\cdot\nabla\phi(x)\right)f\right|\leq\frac{2\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}|\bm{u}|^{2}f.

Here we used (15), (19), (32), and (42). Thus, using all inequalities above in (69), we arrive at the estimate (70). ∎

Next, we compute the term with cubic order of $\bm{u}$ in (70).

Lemma 3.8.

Let $f$ be a solution of (61), and let $\bm{u}$ be given as in (61). Assume (15). Then, we have

(74)

\begin{split}&\left|2\int_{\Omega}\frac{1}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx\right|\\ &\leq\frac{3\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

Proof.

By Using (15), (19), and (32), we have

(75)

\left|2\int_{\Omega}\frac{1}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx\right|\leq\frac{2\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\int_{\Omega}|\bm{u}|^{3}f\,dx.

Next, employing the interpolation inequality (68) and we arrive at:

(76)

\begin{split}\frac{2\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\int_{\Omega}|\bm{u}|^{3}f\,dx&\leq\frac{3\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\\ &\qquad+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

Finally combining (75) with (76) and using $D(x)\geq\Cr{const:D_{M}in}\geq 1$ , we obtain the result (74). ∎

Next, we study the integrals involving $\nabla\phi$ and $\nabla^{2}\phi$ in (69), by using the positive 2nd term. We will show that with large diffusion bound $\Cr{const:D_{M}in}$ , we can reduce the original (69) into a specific form.

Proposition 3.9.

Let $f$ be a solution of (61), and let $\bm{u}$ be given as in (61). Assume the lower bound of the diffusion $D(x)$ , $\Cr{const:D_{M}in}\geq 1$ is large enough such that the following condition holds:

(77)

\max\left\{3\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}},2\Cr{const:log_{f}}\Cr{const:grad_{D}}\|\nabla\phi\|_{L^{\infty}(\Omega)},4(1+n)(\Cr{const:log_{f}}+1)^{2}\Cr{const:grad_{D}}^{2}\right\}\leq\Cr{const:D_{M}in},

where the constant $\Cr{const:log_{f}}$ is the bound of the solution and $\Cr{const:grad_{D}}$ is the bound of the gradient of $D(x)$ , as defined in (32) and (19) in Section 1. Then we obtain,

(78)

\begin{split}\frac{d^{2}F}{dt^{2}}[f](t)&\geq-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx+\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad-\frac{1}{6}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

Remark 3.10.

We again point out the fact that $\Cr{const:log_{f}}$ and $\Cr{const:3.Sobolev}$ are uniformly bounded with respect to $\Cr{const:D_{M}in}$ . The constants $\Cr{const:grad_{D}}$ and $\|\nabla\phi\|_{L^{\infty}(\Omega)}$ are independent of $\Cr{const:D_{M}in}$ , so for given $n$ , $f_{0}$ , $\phi$ , (79), (80), and (81) below hold for sufficiently large $\Cr{const:D_{M}in}$ .

Proof.

Note that the condition (77) yields the following inequalities:

(79)

\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{\Cr{const:D_{M}in}}\leq\frac{1}{3},

(80)

\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\leq\frac{1}{2},

and

(81)

(\Cr{const:log_{f}}+1)^{2}\Cr{const:grad_{D}}^{2}\leq\frac{1}{4(1+n)}\Cr{const:D_{M}in}.

First, combining (70) and (74), we obtain

(82)

\begin{split}\frac{d^{2}F}{dt^{2}}[f](t)&\geq 2\int_{\Omega}[(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})\\ &\qquad\qquad-\biggl{(}\frac{1}{2}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\biggr{)}|\bm{u}|^{2}]f\,dx\\ &\qquad+2\int_{\Omega}\biggl{(}1-\frac{(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}\biggr{)}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad-\frac{3\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad-\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

We consider the integral of $|\bm{u}|^{2}f$ in (82). Using (20), we get,

(83)

(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})\geq-\lambda|\bm{u}|^{2}.

Then applying the assumption (80), we have

(84)

\begin{split}&2\int_{\Omega}\biggl{(}(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})-\Bigl{(}\frac{1}{2}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\Bigr{)}|\bm{u}|^{2}\biggr{)}f\,dx\\ &\quad\geq-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx.\end{split}

Next we consider the integral of $D(x)|\nabla\bm{u}|^{2}f$ in (82). Using (79), we obtain,

(85)

\begin{split}&2\biggl{(}1-\frac{(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}\biggr{)}-\frac{3\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\\ &\quad\geq 2\biggl{(}1-\frac{(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}\biggr{)}-\frac{1}{2}.\end{split}

Thus, by (81) that

(86)

\frac{(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}\leq\frac{1}{4}.

Combining (85) and (86), we arrive at

(87)

\begin{split}2\int_{\Omega}\biggl{(}1&-\frac{(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}\biggr{)}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &-\frac{3\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\geq\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx.\end{split}

Finally, use (79) for the coefficient of the last term in the right-hand side of (82), we have

(88)

\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}\Cr{const:3.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\leq\frac{1}{6}.

With the fact that $\Cr{const:log_{f}}$ and $\Cr{const:3.Sobolev}$ are independent of $\Cr{const:D_{M}in}$ , combining (84), (87), and (88), we obtain (78). ∎

Using Proposition 3.9 and with suitably large $\Cr{const:D_{M}in}$ , we further reduce inequality the dissipation rate functional (78).

Lemma 3.11.

Let $f$ be a solution of (61), and let $\bm{u}$ be given as in (61). Then for a given $\gamma>0$ , there exists a sufficiently large positive constant $\Cr{const:D_{M}in}\geq 1$ which depends only on $n$ , $\|\phi\|_{L^{\infty}(\Omega)}$ , $\|\nabla\phi\|_{L^{\infty}(\Omega)}$ , $\lambda$ from (20)(the lower bound of the Hessian of $\phi$ ), $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14)(the bounds of the initial datum $f_{0}$ ), $\Cr{const:grad_{D}}$ defined in (19)(the bound of the gradient of $D$ ), $\Cr{const:3.Sobolev}$ appeared in (66), and $\gamma$ such that if (15) holds, then,

(89)

\frac{d^{2}}{dt^{2}}F[f](t)\geq\gamma\int_{\Omega}|\bm{u}|^{2}f\,dx-\frac{1}{6}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.

Proof.

By Lemma 3.3 together with (15),

(90)

\begin{split}&-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx+\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\geq-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx+\Cr{const:D_{M}in}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\\ &\geq\left(-2(\lambda+1)+\frac{\Cr{const:D_{M}in}}{\Cr{const:3.Poincare}}\right)\int_{\Omega}|\bm{u}|^{2}f\,dx.\end{split}

Note that $\Cr{const:3.Poincare}$ depends only on $n$ , $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14), $\Cr{const:grad_{D}}$ defined in (19), and $\|\phi\|_{L^{\infty}(\Omega)}$ . Thus, first we take large $\Cr{const:D_{M}in}\geq 1$ such that the assumptions (79), (80), and (81) hold. Next, for $\gamma>0$ , take $\Cr{const:D_{M}in}\geq 1$ further sufficiently large such that

(91)

-2(\lambda+1)+\frac{\Cr{const:D_{M}in}}{\Cr{const:3.Poincare}}\geq\gamma.

Then, we can use (78) and (90), hence we obtain (89). ∎

Next, we will recall the following Gronwall-type inequality from [16].

Lemma 3.12 (Lemma 3.16 in [16]).

Let $c,d,p>0$ be positive constants, such that $p>1$ . Let $g:[0,\infty)\rightarrow\mathbb{R}$ be a non-negative function that satisfies the following differential inequality,

(92)

\frac{dg}{dt}\leq-cg+dg^{p}.

(93)

g(0)<\left(\frac{c}{d}\right)^{\frac{1}{p-1}},

then, we obtain for $t>0$ ,

(94)

g(t)\leq\left(g(0)^{-p+1}-\frac{d}{c}\right)^{-\frac{1}{p-1}}e^{-ct}.

This Gronwall-type inequality will allow us to show the exponential decay of the dissipation rate of the free energy in (63), the main result of this Section.

Proof of Theorem 3.1.

For any $\gamma>0$ , using Lemma 3.11, take sufficiently large positive number $\Cr{const:D_{M}in}\geq 1$ . Then with specifically the following quantities, (89) becomes (92):

g(t)=-\frac{d}{dt}F[f](t)=\int_{\Omega}|\bm{u}|^{2}f\,dx,\quad c=\gamma,\quad p=3,\quad\text{and}\quad d=\frac{1}{6}.

Thus, if

(95)

g(0)=\int_{\Omega}|\nabla(D(x)\log f_{0}(x)+\phi(x))|^{2}f_{0}\,dx<\left(6\gamma\right)^{\frac{1}{2}},

then by Lemma 3.12

(96)

\begin{split}g(t)&=\int_{\Omega}|\bm{u}|^{2}f\,dx\\ &\leq\left(\left(\int_{\Omega}|\nabla(D(x)\log f_{0}(x)+\phi(x))|^{2}f_{0}\,dx\right)^{-2}-\frac{1}{6\gamma}\right)^{-\frac{1}{2}}e^{-\gamma t}.\end{split}

Taking $\Cr{const:3.Initial_{E}nergy}=(6\gamma)^{\frac{1}{2}}$ and

(97)

\Cr{const:3.Exponential_{C}oefficient}=\left(\left(\int_{\Omega}|\nabla(D(x)\log f_{0}(x)+\phi(x))|^{2}f_{0}\,dx\right)^{-2}-\frac{1}{6\gamma}\right)^{-\frac{1}{2}},

we will arrive at the conclusion of Theorem 3.1. ∎

In this Section, we had demonstrated the exponential decay of the time derivative of the free energy in the case that $D(x)$ is inhomogeneous and the mobility is constant. In the next section, we will consider the case of inhomogeneous diffusion $D(x)$ and variable mobility $\pi(x,t)$ .

4. Inhomogeneous diffusion case with variable mobility

This section will be devoted to the following nonlinear Fokker-Planck equation with inhomogeneity in both diffusion $D(x)$ and mobility $\pi(x,t)$ , which are bounded periodic positive functions defined in a bounded domain $\Omega$ in the Euclidean space of $n$ -dimension.

(98)

\left\{\begin{aligned} \frac{\partial f}{\partial t}&+\operatorname{div}\left(f\bm{u}\right)=0,&\quad&x\in\Omega,\quad t>0,\\ \bm{u}&=-\frac{1}{\pi(x,t)}\nabla\left(D(x)\log f+\phi(x)\right),&\quad&x\in\Omega,\quad t>0,\\ f(&x,0)=f_{0}(x),&\quad&x\in\Omega.\end{aligned}\right.

Again, without loss of generality, we take $\Omega=[0,1)^{n}\subset\mathbb{R}^{n}$ . For the convenience of the readers, we recall (as defined in Section 1) that the periodic function $D(x)$ is bounded from below with the constant $\Cr{const:D_{M}in}\geq 1$ , and the periodic function $\pi(x,t)$ is bounded both from below and above by the positive constants $\Cr{const:Pi_{M}in}$ and $\Cr{const:Pi_{M}ax}$ , namely

D(x)\geq\Cr{const:D_{M}in},\quad\Cr{const:Pi_{M}in}\leq\pi(x,t)\leq\Cr{const:Pi_{M}ax}

for any $x\in\Omega$ and $t>0$ .

The free energy $F$ and the basic energy law (9) still takes the standard form:

(99)

F[f]:=\int_{\Omega}\left(D(x)f(\log f-1)+f\phi(x)\right)\,dx,

and the system satisfies the following energy dissipation law:

(100)

\frac{dF}{dt}[f](t)=-\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx=:-D_{\mathrm{dis}}[f](t).

We will first state the main result of this section: for a given system in (98), with suitable conditions of the initial data and the mobility, under relatively mild inhomogeneity conditions, one can find a diffusion $D(x)$ that is large enough, such that the system will convergence exponentially to a equilibrium.

Theorem 4.1.

Assume $n=1,2,3$ . For a fixed constant $\gamma>0$ , there exist positive constants $\Cr{const:D_{M}in},$ $\Cr{const:Pi_{T}ime},$ $\Cr{const:grad_{P}i}$ and $\Cl{const:4.Initial_{E}nergy}$ , which depend only on the given constant $\gamma$ , $n$ , the potential $\phi$ (in terms of Hessian bound $\lambda$ defined in (20), $\|\phi\|_{L^{\infty}(\Omega)}$ , and $\|\nabla\phi\|_{L^{\infty}(\Omega)}$ ), the bound of initial data $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14), the bound $\Cr{const:grad_{D}}$ of $\nabla D(x)$ defined in (19), and the bounds for the mobility $\Cr{const:Pi_{M}in}$ , $\Cr{const:Pi_{M}ax}$ defined in (16), such that if (15), (17), (18) hold and,

(101)

\int_{\Omega}\pi(x,0)|\nabla D\log f_{0}+\phi(x)|^{2}\,dx\leq\Cr{const:4.Initial_{E}nergy},

then for $t>0$ , the following is true,

(102)

\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx\leq\Cr{const:4.Exponential_{C}oefficient}e^{-\gamma t},

for a positive constant $\Cl{const:4.Exponential_{C}oefficient}>0$ .

Remark 4.2.

Unlike Sections 2 and 3, the velocity $\bm{u}$ in this section is not a gradient field, which means that $\overline{\bm{u}}\neq 0$ . We will need to re-evaluate those estimates obtained in the previous sections and if necessary, derive the ones under the new conditions. Here we start with the Sobolev-type inequality for $\bm{u}$ with the mean value $\overline{\bm{u}}$ .

Lemma 4.3.

Let $n$ be a natural number, for $\varepsilon>0$ , the positive constants $\Cr{const:Pi_{M}in},\Cr{const:grad_{P}i}$ being defined as the bounds of the mobility (16) and the bound of the derivative of the mobility (18) respectively. Let $f$ be a solution of (98), and let $\bm{u}$ be given as in (98). If

(103)

\Cr{const:grad_{P}i}\leq\Cr{const:Pi_{M}in}\left(\frac{\varepsilon}{2}\right)^{\frac{1}{2}},

then there exists $\Cl{const:4.Sobolev}>0$ which depends only on $n$ , $\pi$ (in terms of $\Cr{const:Pi_{M}in}$ , $\Cr{const:Pi_{M}ax}$ defined in (16)), $f_{0}$ (in terms of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined on (14)), the gradient of $D(x)$ (in terms of $\Cr{const:grad_{D}}$ defined in (19)), and $\|\phi\|_{L^{\infty}(\Omega)}$ such that,

(104)

\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\leq\Cr{const:4.Sobolev}\left(\int_{\Omega}\left(2|\nabla\bm{u}|^{2}+\varepsilon|\bm{u}|^{2}\right)f\,dx\right)^{\frac{1}{2}},

where $2<p^{*}<\infty$ for $n=1,2$ , and $\frac{1}{p^{*}}=\frac{1}{2}-\frac{1}{n}$ for $n\geq 3$ .

Proof.

First, using the definition of $\Cr{const:Pi_{M}in}$ , the positive bounds for the mobility (16), we have,

(105)

\begin{split}\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}&=\left(\int_{\Omega}\left|\frac{1}{\pi(x,t)}\nabla\left(D(x)\log f+\phi(x)\right)\right|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\\ &\leq\frac{1}{\Cr{const:Pi_{M}in}}\left(\int_{\Omega}\left|\nabla\left(D(x)\log f+\phi(x)\right)\right|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}.\end{split}

Then, we can use Lemma 3.2 that

(106)

\begin{split}&\left(\int_{\Omega}\left|\nabla\left(D(x)\log f+\phi(x)\right)\right|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\\ &\quad\leq\Cr{const:3.Sobolev}\left(\int_{\Omega}|\nabla^{2}\left(D(x)\log f+\phi(x)\right)|^{2}f\,dx\right)^{\frac{1}{2}}.\end{split}

Using the definition of $\Cr{const:Pi_{M}in}$ and $\Cr{const:Pi_{M}ax}$ in the positive bounds for the mobility (16), we obtain

(107)

\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\leq\frac{\Cr{const:Pi_{M}ax}\Cr{const:3.Sobolev}}{\Cr{const:Pi_{M}in}}\left(\int_{\Omega}\left|-\frac{1}{\pi(x,t)}\nabla^{2}\left(D(x)\log f+\phi(x)\right)\right|^{2}f\,dx\right)^{\frac{1}{2}}.

Here

(108)

\begin{split}\nabla\bm{u}&=-\frac{1}{\pi(x,t)}\nabla^{2}\left(D(x)\log f+\phi(x)\right)\\ &\quad+\frac{1}{\pi^{2}(x,t)}\nabla\pi(x,t)\otimes\nabla\left(D(x)\log f+\phi(x)\right).\end{split}

Thus, we obtain by the Young inequality and the definition of $\bm{u}$ that,

(109)

\begin{split}&\left|-\frac{1}{\pi(x,t)}\nabla^{2}\left(D(x)\log f+\phi(x)\right)\right|^{2}\\ &\leq 2|\nabla\bm{u}|^{2}+2\left|\frac{1}{\pi^{2}(x,t)}\nabla\pi(x,t)\otimes\nabla\left(D(x)\log f+\phi(x)\right)\right|^{2}\\ &=2|\nabla\bm{u}|^{2}+2\left|\frac{1}{\pi(x,t)}\nabla\pi(x,t)\otimes\bm{u}\right|^{2}\\ &=2|\nabla\bm{u}|^{2}+\frac{2}{\pi^{2}(x,t)}\left|\nabla\pi(x,t)\right|^{2}\left|\bm{u}\right|^{2}.\end{split}

Therefore, we arrive at

\begin{split}&\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\\ &\quad\leq\frac{\Cr{const:Pi_{M}ax}\Cr{const:3.Sobolev}}{\Cr{const:Pi_{M}in}}\left(\int_{\Omega}\left(2|\nabla\bm{u}|^{2}+\frac{2}{\pi^{2}(x,t)}\left|\nabla\pi(x,t)\right|^{2}\left|\bm{u}\right|^{2}\right)f\,dx\right)^{\frac{1}{2}}.\end{split}

Finally, we compute the coefficient of $|\bm{u}|^{2}$ in the right hand side. Using (103), the positive bounds for the mobility (16), and the bound for the derivatives of the mobility (18), we obtain

(110)

\frac{1}{\pi^{2}(x,t)}\left|\nabla\pi(x,t)\right|^{2}\leq\frac{\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}\leq\frac{\varepsilon}{2},

hence we obtain (104) by taking $\Cr{const:4.Sobolev}=\frac{\Cr{const:Pi_{M}ax}\Cr{const:3.Sobolev}}{\Cr{const:Pi_{M}in}}$ . ∎

Remark 4.4.

We emphasize that the Sobolev-type constant $\Cr{const:4.Sobolev}$ in the above lemma is independent of $\Cr{const:grad_{P}i}$ , hence the gradient of the mobility $\pi$ .

With this result, next we will show that if the bounds for the gradient mobility $\Cr{const:grad_{P}i}$ , defined in (18), is not too large, then we can obtain the Poincare type inequality for $\bm{u}$ .

Lemma 4.5.

Let $f$ be a solution of (98), and let $\bm{u}$ be given as in (98). Assume the bound for the gradient mobility $\Cr{const:grad_{P}i}$ , defined in (18) satisfies the following condition:

(111)

\Cr{const:grad_{P}i}\leq\frac{\Cr{const:Pi_{M}in}}{2\Cr{const:4.Sobolev}},

where $\Cr{const:Pi_{M}in}$ is the lower bound of $\pi$ appeared in (16), $\Cr{const:4.Sobolev}$ is appeared in (104). Then, there is a constant $\Cl{const:4.Poincare}>0$ which depends only on $n$ , $\pi$ (in terms of $\Cr{const:Pi_{M}in}$ , $\Cr{const:Pi_{M}ax}$ defined in (16)), $f_{0}$ (in terms of $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14)), the gradient of $D(x)$ (in terms of $\Cr{const:grad_{D}}$ defined in (19)), and $\|\phi\|_{L^{\infty}(\Omega)}$ such that:

(112)

\int_{\Omega}|\bm{u}|^{2}f\,dx\leq\Cr{const:4.Poincare}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx.

Remark 4.6.

Note from the proof of Lemma 4.3, that $\Cr{const:4.Sobolev}$ is taken as $\Cr{const:4.Sobolev}=\frac{\Cr{const:Pi_{M}ax}\Cr{const:3.Sobolev}}{\Cr{const:Pi_{M}in}}$ . Thus, the condition (111) can be written as $\Cr{const:grad_{P}i}\leq\frac{\Cr{const:Pi_{M}in}^{2}}{2\Cr{const:Pi_{M}ax}\Cr{const:3.Sobolev}}$ .

Proof.

By the Hölder inequality and (7), we have

(113)

\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{\frac{1}{2}}\leq\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}.

We choose $\varepsilon>0$ later and assume (103). Then, by the Sobolev type inequality (104), we obtain

(114)

\left(\int_{\Omega}|\bm{u}|^{p^{*}}f\,dx\right)^{\frac{1}{p^{*}}}\leq\Cr{const:4.Sobolev}\left(\int_{\Omega}\left(2|\nabla\bm{u}|^{2}+\varepsilon|\bm{u}|^{2}\right)f\,dx\right)^{\frac{1}{2}}.

Now we choose $\varepsilon$ as

(115)

\Cr{const:4.Sobolev}^{2}\varepsilon=\frac{1}{2},

and take $p^{*}=2.$ Then, we obtain

(116)

\frac{1}{2}\int_{\Omega}|\bm{u}|^{2}f\,dx\leq 2\Cr{const:4.Sobolev}^{2}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx.

Plugging (115) into (103), we get (111). Taking $\Cr{const:4.Poincare}=4\Cr{const:4.Sobolev}^{2}$ , we obtain (112). ∎

In contrast to Lemma 3.2, the Sobolev-type of inequality (104) has to include an extra quadratic term of $\bm{u}$ in the right hand side. Here we will re-derive the interpolation inequality like that in Lemma 3.5.

Lemma 4.7.

Let $n=1,2,3$ and let $\varepsilon>0$ , and let $\Cr{const:Pi_{M}in},\Cr{const:grad_{P}i}>0$ be the constants defined in the bounds for the mobility (16) and the bound of the derivative of the mobility (18) respectively. Let $f$ be a solution of (98), and let $\bm{u}$ be given as in (98). Under the condition:

(117)

\Cr{const:grad_{P}i}\leq\Cr{const:Pi_{M}in},

we will have the following estimate:

(118)

\begin{split}\int_{\Omega}|\bm{u}|^{3}f\,dx&\leq\frac{3}{2}\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx+\frac{3}{2}\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}|\bm{u}|^{2}f\,dx\\ &\quad+\frac{1}{4}\Cr{const:4.Sobolev}^{\frac{3}{2}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3},\end{split}

where $\Cr{const:4.Sobolev}$ is the constant defined in the Sobolev estimate (104) in Lemma 4.3.

Proof.

We consider the case $n=3$ first. Let $\alpha,\beta>0$ be constants satisfying $\alpha+\beta=1$ , and let $q>1$ be an exponent. Then, by the Hölder’s inequality,

(119)

\int_{\Omega}|\bm{u}|^{3}f\,dx\leq\left(\int_{\Omega}|\bm{u}|^{3\alpha q}f\,dx\right)^{\frac{1}{q}}\left(\int_{\Omega}|\bm{u}|^{3\beta q^{\prime}}f\,dx\right)^{\frac{1}{q^{\prime}}},

where $q^{\prime}$ is the Hölder’s conjugate, namely, $\frac{1}{q}+\frac{1}{q^{\prime}}=1$ . Next, we put a constraint, $3\alpha q=p^{\ast}$ , in order to apply the Sobolev type inequality (104) with $\varepsilon=2$ . Note that (117) guarantees the assumption (103) in Lemma 4.3. Then (119) turns into

(120)

\int_{\Omega}|\bm{u}|^{3}f\,dx\leq\Cr{const:4.Sobolev}^{\frac{p^{\ast}}{q}}\left(\int_{\Omega}2(|\nabla\bm{u}|^{2}+|\bm{u}|^{2})f\,dx\right)^{\frac{p^{\ast}}{2q}}\left(\int_{\Omega}|\bm{u}|^{3\beta q^{\prime}}f\,dx\right)^{\frac{1}{q^{\prime}}}.

Next, we set other constraints, $3\beta q^{\prime}=2$ and $\frac{p^{\ast}}{2q}<1$ . The Young’s inequality implies,

(121)

\begin{split}&\left(\int_{\Omega}2(|\nabla\bm{u}|^{2}+|\bm{u}|^{2})f\,dx\right)^{\frac{p^{\ast}}{2q}}\left(\int_{\Omega}|\bm{u}|^{3\beta q^{\prime}}f\,dx\right)^{\frac{1}{q^{\prime}}}\\ &\quad\leq\frac{p^{\ast}}{q}\int_{\Omega}(|\nabla\bm{u}|^{2}+|\bm{u}|^{2})f\,dx+\left(1-\frac{p^{\ast}}{2q}\right)\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{\frac{1}{q^{\prime}}\left(1-\frac{p^{\ast}}{2q}\right)^{-1}},\end{split}

which yields:

(122)

\begin{split}\int_{\Omega}|\bm{u}|^{3}f\,dx&\leq\Cr{const:4.Sobolev}^{\frac{p^{\ast}}{q}}\frac{p^{*}}{q}\int_{\Omega}(|\nabla\bm{u}|^{2}+|\bm{u}|^{2})f\,dx\\ &\qquad+\Cr{const:4.Sobolev}^{\frac{p^{\ast}}{q}}\left(1-\frac{p^{\ast}}{2q}\right)\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{\frac{1}{q^{\prime}}\left(1-\frac{p^{\ast}}{2q}\right)^{-1}}.\end{split}

Now we can come back to examine the constraints. Since $3\alpha q=p^{\ast}$ , $3\beta q^{\prime}=2$ , $\alpha+\beta=1$ , and the properties of $q^{\prime}$ , $p^{\ast}$ imply that, $\frac{3\beta}{2}=\frac{1}{q^{\prime}}=1-\frac{3\alpha}{p^{\ast}}=1-\frac{3\alpha}{2}+\frac{3\alpha}{n}.$ From these, one can deduce that $\alpha=\frac{n}{6}=\frac{1}{2}$ , and in turns, $\beta=\frac{1}{2}$ , $q=4$ and $p^{*}=6$ . Plugging these values in (122) and we can obtain (118) directly.

For the case $n=1,2$ , we can take $p^{\ast}=6$ , the same as in the case $n=3$ . Then it is easy to verify that by taking $\alpha=\beta=\frac{1}{2}$ and $q=4$ in (122), one can obtain the inequality (118). ∎

Next we recall the following higher order energy law of (98), which can be derived by direct computations as we had done in [16].

Lemma 4.8 ([16, Proposition 4.15]).

Let $f$ be a solution of (98), and let $\bm{u}$ be given as in (98). Then, we have the following energy law,

(123)

\begin{split}\frac{d^{2}}{dt^{2}}F[f](t)&=2\int_{\Omega}((\nabla^{2}\phi(x))\bm{u}\cdot\bm{u})f\,dx+2\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad-\int_{\Omega}(\log f-1)\nabla|\bm{u}|^{2}\cdot\nabla D(x)f\,dx\\ &\qquad-2\int_{\Omega}(1+\log f)\bm{u}\cdot\nabla D(x)\operatorname{div}\bm{u}f\,dx\\ &\qquad+2\int_{\Omega}\frac{\pi(x,t)}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx\\ &\qquad+2\int_{\Omega}\frac{1}{D(x)}(\log f)^{2}\left(\bm{u}\cdot\nabla D(x)\right)^{2}f\,dx\\ &\qquad+2\int_{\Omega}\frac{1}{D(x)}\log f\left(\bm{u}\cdot\nabla D(x)\right)\left(\bm{u}\cdot\nabla\phi(x)\right)f\,dx\\ &\qquad+\int_{\Omega}\pi_{t}(x,t)|\bm{u}|^{2}f\,dx+\int_{\Omega}|\bm{u}|^{2}\bm{u}\cdot\nabla\pi(x,t)f\,dx\\ &\qquad-2\int_{\Omega}(\log f-1)\frac{1}{\pi(x,t)}|\bm{u}|^{2}\nabla\pi(x,t)\cdot\nabla D(x)f\,dx\\ &\qquad+2\int_{\Omega}(\log f-1)\frac{1}{\pi(x,t)}(\bm{u}\cdot\nabla\pi(x,t))(\bm{u}\cdot\nabla D(x))f\,dx\\ &\qquad+\int_{\Omega}\frac{D(x)}{\pi(x,t)}((\nabla|\bm{u}|^{2})\cdot\nabla\pi(x,t))f\,dx\\ &\qquad-2\int_{\Omega}\frac{D(x)}{\pi(x,t)}((\nabla\bm{u})\bm{u}\cdot\nabla\pi(x,t))f\,dx.\end{split}

We will first derive the following inequality from the second derivative of the free energy above (123).

Lemma 4.9.

Let $f$ be a solution of (98), and let $\bm{u}$ be given as in (98). Then, we have

(124)

\begin{split}&\frac{d^{2}}{dt^{2}}F[f](t)\\ &\geq 2\int_{\Omega}\biggl{(}(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})+\frac{1}{2}\pi_{t}(x,t)|\bm{u}|^{2}\\ &\qquad-\Bigl{(}\frac{1}{2}+\frac{2(\Cr{const:log_{f}}+1)\Cr{const:grad_{D}}\Cr{const:grad_{P}i}}{\Cr{const:Pi_{M}in}}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\Bigr{)}|\bm{u}|^{2}\biggr{)}f\,dx\\ &\quad+2\int_{\Omega}\biggl{(}1-\frac{2(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}-\frac{4D(x)\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}\biggr{)}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\quad+2\int_{\Omega}\frac{\pi(x,t)}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx\\ &\quad+\int_{\Omega}|\bm{u}|^{2}\bm{u}\cdot\nabla\pi(x,t)f\,dx,\end{split}

where $\Cr{const:D_{M}in}$ is the lower bound of $D(x)$ defined in (15), $\Cr{const:Pi_{M}in}$ is the lower bound of $\pi(x,t)$ defined in (16), $\Cr{const:grad_{P}i}$ is the upper bound for the estimate of the gradient of $\pi(x,t)$ defined in (18), $\Cr{const:grad_{D}}$ is the upper bound for the estimate of the gradient of $D(x)$ defined in (19), and $\Cr{const:log_{f}}$ is the bound for the estimate of $\log f$ defined in (32).

Proof.

We start with the estimates of the integrands for the 3rd, 4th, and 7th integrals of (123). By Cauchy’s inequality with $\varepsilon=1/8$ (See [22, p.662]) and the definition of $\Cr{const:D_{M}in}$ , we will get

(125)

\begin{split}|(\log f-1)\nabla|\bm{u}|^{2}\cdot\nabla D(x)f|&\leq 4(|\log f|+1)^{2}|\nabla\bm{u}|^{2}|\nabla D(x)|^{2}f+\frac{1}{4}|\bm{u}|^{2}f\\ &\leq\frac{4\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}D(x)|\nabla\bm{u}|^{2}f+\frac{1}{4}|\bm{u}|^{2}f,\end{split}

(126)

\begin{split}|2(1+\log f)\bm{u}\cdot\nabla D(x)\operatorname{div}\bm{u}f|&\leq 4n(|\log f|+1)^{2}|\nabla\bm{u}|^{2}|\nabla D(x)|^{2}f+\frac{1}{4}|\bm{u}|^{2}f\\ &\leq\frac{4n\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}D(x)|\nabla\bm{u}|^{2}f+\frac{1}{4}|\bm{u}|^{2}f,\end{split}

and

(127)

\left|\frac{2}{D(x)}\log f\left(\bm{u}\cdot\nabla D(x)\right)\left(\bm{u}\cdot\nabla\phi(x)\right)f\right|\leq\frac{2\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}|\bm{u}|^{2}f.

Next we estimate the 10th, 11st, 12nd and 13rd terms of the right-hand side of (123). Using the Cauchy-Schwarz inequality that

(128)

\left|2(\log f-1)\frac{1}{\pi(x,t)}|\bm{u}|^{2}\nabla\pi(x,t)\cdot\nabla D(x)f\right|\leq\frac{2(\Cr{const:log_{f}}+1)\Cr{const:grad_{D}}\Cr{const:grad_{P}i}}{\Cr{const:Pi_{M}in}}|\bm{u}|^{2}f,

(129)

\left|2(\log f-1)\frac{1}{\pi(x,t)}(\bm{u}\cdot\nabla\pi(x,t))(\bm{u}\cdot\nabla D(x))f\right|\leq\frac{2(\Cr{const:log_{f}}+1)\Cr{const:grad_{D}}\Cr{const:grad_{P}i}}{\Cr{const:Pi_{M}in}}|\bm{u}|^{2}f,

(130)

\begin{split}\left|\frac{D(x)}{\pi(x,t)}((\nabla|\bm{u}|^{2})\cdot\nabla\pi(x,t))f\right|&\leq\frac{2D(x)}{\pi(x,t)}|\nabla\bm{u}||\bm{u}||\nabla\pi(x,t)|f\\ &\leq\frac{4(D(x))^{2}\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}|\nabla\bm{u}|^{2}f+\frac{1}{4}|\bm{u}|^{2}f,\end{split}

and

(131)

\begin{split}\left|2\frac{D(x)}{\pi(x,t)}((\nabla\bm{u})\bm{u}\cdot\nabla\pi(x,t))f\right|&\leq\frac{2D(x)}{\pi(x,t)}|\nabla\bm{u}||\bm{u}||\nabla\pi(x,t)|f\\ &\leq\frac{4(D(x))^{2}\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}|\nabla\bm{u}|^{2}f+\frac{1}{4}|\bm{u}|^{2}f.\end{split}

Combining all inequalities above, ignoring the integral of $\frac{1}{D(x)}(\log f)^{2}(\bm{u}\cdot\nabla D(x))^{2}f$ , and use them in (123), we arrive at the reduced estimate (124). ∎

Next we consider the terms involving cubic order of $\bm{u}$ in (124).

Lemma 4.10.

Let $f$ be a solution of (98), and let $\bm{u}$ be given as in (98). Assume (117) and (15). Then, we have,

(132)

\begin{split}&2\left|\int_{\Omega}\frac{\pi(x,t)}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx\right|+\left|\int_{\Omega}|\bm{u}|^{2}\bm{u}\cdot\nabla\pi(x,t)f\,dx\right|\\ &\leq\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad+\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}|\bm{u}|^{2}f\,dx\\ &\qquad+\left(\frac{\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{2\Cr{const:D_{M}in}}+\frac{\Cr{const:grad_{P}i}}{4}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3},\end{split}

where $\Cr{const:D_{M}in}$ is the lower bound of $D(x)$ defined in (15), $\Cr{const:Pi_{M}ax}$ is the upper bound of $\pi(x,t)$ defined in (16), $\Cr{const:grad_{P}i}$ is the upper bound for the estimate of the gradient of $\pi(x,t)$ defined in (18), $\Cr{const:grad_{D}}$ is the upper bound for the estimate of the gradient of $D(x)$ defined in (19), $\Cr{const:log_{f}}$ is the bound for the estimate of $\log f$ defined in (32), and $\Cr{const:4.Sobolev}$ is appeared in (104).

Proof.

First, we compute the first term of the left hand side of (132). Using (118) and (15), we compute

(133)

\begin{split}&\left|2\int_{\Omega}\frac{\pi(x,t)}{D(x)}|\bm{u}|^{2}\log f\left(\bm{u}\cdot\nabla D(x)\right)f\,dx\right|\leq\frac{2\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\int_{\Omega}|\bm{u}|^{3}f\,dx\\ &\leq\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{\Cr{const:D_{M}in}}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\\ &\qquad+\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{\Cr{const:D_{M}in}}\int_{\Omega}|\bm{u}|^{2}f\,dx+\frac{\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}\\ &\leq\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{\Cr{const:D_{M}in}}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad+\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{\Cr{const:D_{M}in}}\int_{\Omega}|\bm{u}|^{2}f\,dx+\frac{\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{2\Cr{const:D_{M}in}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

Next, we consider the second term of the left hand side of (132). Using (118) and (15) again that,

(134)

\begin{split}&\left|\int_{\Omega}|\bm{u}|^{2}\bm{u}\cdot\nabla\pi(x,t)f\,dx\right|\leq\Cr{const:grad_{P}i}\int_{\Omega}|\bm{u}|^{3}f\,dx\\ &\leq\frac{3\Cr{const:grad_{P}i}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{2}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx+\frac{3\Cr{const:grad_{P}i}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{2}\int_{\Omega}|\bm{u}|^{2}f\,dx+\frac{\Cr{const:grad_{P}i}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{4}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}\\ &\leq\frac{3\Cr{const:grad_{P}i}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{2}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx+\frac{3\Cr{const:grad_{P}i}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{2}\int_{\Omega}|\bm{u}|^{2}f\,dx\\ &\qquad+\frac{\Cr{const:grad_{P}i}\Cr{const:4.Sobolev}^{\frac{3}{2}}}{4}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

Combining (133) and (134), we obtain (132). ∎

Now we are ready to demonstrate that under the assumptions on suitable sufficient “smallness” conditions for the terms involving the gradient of the mobility $|\nabla\pi|$ , and $|\pi_{t}|$ , in terms of $\Cr{const:grad_{P}i}$ and $\Cr{const:Pi_{T}ime}$ , together with the assumption of $\Cr{const:D_{M}in}$ being sufficiently large (the lower bound of the diffusion), one can deduce from (123) the following proposition.

Proposition 4.11.

Let $f$ be a smooth classical solution of (98), and let $\bm{u}$ be given as in (98). Assume the lower bound of the diffusion $D(x)$ , $\Cr{const:D_{M}in}\geq 1$ , is large enough and the upper bounds of $|\pi_{t}|$ and $|\nabla\pi|$ , in terms of $\Cr{const:Pi_{T}ime}\leq\frac{1}{6}$ and $\Cr{const:grad_{P}i}$ , are small enough such that the following conditions hold:

(135)

\begin{split}\max\biggl{\{}&12\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}\Cr{const:4.Sobolev}^{\frac{3}{2}},12\Cr{const:log_{f}}\Cr{const:grad_{D}}\|\nabla\phi\|_{L^{\infty}(\Omega)},\\ &16(1+n)(\Cr{const:log_{f}}+1)^{2}\Cr{const:grad_{D}}^{2}\biggr{\}}\leq\Cr{const:D_{M}in},\end{split}

and

(136)

\Cr{const:grad_{P}i}\leq\min\left\{\frac{1}{6\Cr{const:4.Sobolev}^{\frac{3}{2}}},\frac{\Cr{const:Pi_{M}in}}{24(\Cr{const:log_{f}}+1)\Cr{const:grad_{D}}},\frac{\Cr{const:Pi_{M}in}}{4\sqrt{2(\sqrt{n}\Cr{const:grad_{D}}+1)\Cr{const:D_{M}in}}},\Cr{const:Pi_{M}in}\right\}.

Here, $\Cr{const:D_{M}in}$ is the lower bound of $D(x)$ defined in (15), $\Cr{const:Pi_{M}in}$ and $\Cr{const:Pi_{M}ax}$ are the lower and the upper bounds of $\pi(x,t)$ defined in (16) respectively. The bound $\Cr{const:Pi_{T}ime}$ is the bound for the estimate of the time derivative of $\pi$ defined in (17), $\Cr{const:grad_{P}i}$ is the upper bound for the estimate of the gradient of $\pi(x,t)$ defined in (18), $\Cr{const:grad_{D}}$ is the upper bound for the estimate of the gradient of $D(x)$ defined in (19), $\Cr{const:log_{f}}$ is the bound for the estimate of $\log f$ defined in (32), and $\Cr{const:4.Sobolev}$ is appeared in (104). Then, we obtain,

(137)

\begin{split}\frac{d^{2}}{dt^{2}}F[f](t)&\geq-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx+\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\quad-\frac{1}{12}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

Remark 4.12.

We emphasize that all of the constants in the left-hand side of (135) are independent of $\Cr{const:D_{M}in}$ and $\Cr{const:grad_{P}i}$ , and every constant in the right-hand side of (136) except for $\Cr{const:D_{M}in}$ is independent of $\Cr{const:grad_{P}i}$ . Thus, in order to guarantee (135) and (136), first we fix sufficiently small $\Cr{const:D_{M}in}$ so (135) holds. Next, for the fixed $\Cr{const:D_{M}in}$ , we take sufficiently large $\Cr{const:grad_{P}i}$ to ensure the inequality (136).

Proof.

Note that the conditions (135), $\Cr{const:Pi_{T}ime}\leq\frac{1}{6}$ , and (136) yield the following inequalities:

(138)

\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\leq\frac{1}{2},

(139)

\frac{1}{2}\Cr{const:Pi_{T}ime}+\frac{2(\Cr{const:log_{f}}+1)\Cr{const:grad_{P}i}\Cr{const:grad_{D}}}{\Cr{const:Pi_{M}in}}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\leq\frac{1}{4},

(140)

(\Cr{const:log_{f}}+1)^{2}\Cr{const:grad_{D}}^{2}\leq\frac{1}{16(1+n)}\Cr{const:D_{M}in},

(141)

(\sqrt{n}\Cr{const:grad_{D}}+1)\Cr{const:D_{M}in}\Cr{const:grad_{P}i}^{2}\leq\frac{1}{32}\Cr{const:Pi_{M}in}^{2},

and $\Cr{const:grad_{P}i}\leq\Cr{const:Pi_{M}in}.$

First, combining (124) and (132), we obtain

(142)

\begin{split}\frac{d^{2}}{dt^{2}}F[f](t)&\geq 2\int_{\Omega}\biggl{(}(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})+\frac{1}{2}\pi_{t}(x,t)|\bm{u}|^{2}\\ &\quad-\Bigl{(}\frac{1}{2}+\frac{2(\Cr{const:log_{f}}+1)\Cr{const:grad_{D}}\Cr{const:grad_{P}i}}{\Cr{const:Pi_{M}in}}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\Bigr{)}|\bm{u}|^{2}\biggr{)}f\,dx\\ &\quad+2\int_{\Omega}\biggl{(}1-\frac{2(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}\\ &\qquad\qquad\qquad-\frac{4D(x)\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}\biggr{)}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\quad-\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\quad-\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}|\bm{u}|^{2}f\,dx\\ &\quad-\left(\frac{\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{2\Cr{const:D_{M}in}}+\frac{\Cr{const:grad_{P}i}}{4}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.\end{split}

We consider the integral of $|\bm{u}|^{2}f$ in (142). Using (20) and (17), we get

(143)

(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})+\frac{1}{2}\pi_{t}(x,t)|\bm{u}|^{2}\geq\left(-\lambda-\frac{1}{2}\Cr{const:Pi_{T}ime}\right)|\bm{u}|^{2}.

Then applying the assumptions (138) and (139), we have

(144)

\begin{split}&2\int_{\Omega}\biggl{(}(\nabla^{2}\phi(x)\bm{u}\cdot\bm{u})+\frac{1}{2}\pi_{t}(x,t)|\bm{u}|^{2}\\ &\qquad-\Bigl{(}\frac{1}{2}+\frac{2(\Cr{const:log_{f}}+1)\Cr{const:grad_{D}}\Cr{const:grad_{P}i}}{\Cr{const:Pi_{M}in}}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\Bigr{)}|\bm{u}|^{2}\biggr{)}f\,dx\\ &\qquad-\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}|\bm{u}|^{2}f\,dx\\ &\geq-2\biggl{(}\lambda+\frac{1}{2}\Cr{const:Pi_{T}ime}+\frac{1}{2}+\frac{2(\Cr{const:log_{f}}+1)\Cr{const:grad_{D}}\Cr{const:grad_{P}i}}{\Cr{const:Pi_{M}in}}+\frac{\Cr{const:log_{f}}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}\|\nabla\phi\|_{L^{\infty}(\Omega)}\\ &\qquad\quad+\frac{1}{2}\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\biggr{)}\int_{\Omega}|\bm{u}|^{2}f\,dx\\ &\geq-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx.\end{split}

Next we consider the integrals in (142) that contain term with $D(x)|\nabla\bm{u}|^{2}f$ . Note, using (138), we obtain

(145)

\begin{split}&2\biggl{(}1-\frac{2(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}-\frac{4D(x)\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}\biggr{)}\\ &\qquad-\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\\ &\geq 2\biggl{(}1-\frac{2(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}-\frac{4\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}\|D\|_{L^{\infty}(\Omega)}\biggr{)}-\frac{1}{2}.\end{split}

Here we use (21) and $\Cr{const:D_{M}in}\geq 1$ , then

(146)

\|D\|_{L^{\infty}(\Omega)}\leq\Cr{const:D_{M}in}+\sqrt{n}\Cr{const:grad_{D}}\leq\left(1+\sqrt{n}\Cr{const:grad_{D}}\right)\Cr{const:D_{M}in}.

Thus, by the assumptions of (140) and (141), we have that

(147)

\frac{2(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}+\frac{4\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}\|D\|_{L^{\infty}(\Omega)}\leq\frac{1}{4}.

Combining (145) and (147), we arrive at

(148)

\begin{split}&\quad 2\int_{\Omega}\biggl{(}1-\frac{2(1+n)\Cr{const:grad_{D}}^{2}(\Cr{const:log_{f}}+1)^{2}}{\Cr{const:D_{M}in}}-\frac{4D(x)\Cr{const:grad_{P}i}^{2}}{\Cr{const:Pi_{M}in}^{2}}\biggr{)}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\qquad-\left(\frac{3\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{\Cr{const:D_{M}in}}+\frac{3\Cr{const:grad_{P}i}}{2}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\geq\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx.\end{split}

Finally, we use (138) to estimate the coefficient of the last term (the coefficient of the integral cubed), and we have that,

(149)

\left(\frac{\Cr{const:log_{f}}\Cr{const:Pi_{M}ax}\Cr{const:grad_{D}}}{2\Cr{const:D_{M}in}}+\frac{\Cr{const:grad_{P}i}}{4}\right)\Cr{const:4.Sobolev}^{\frac{3}{2}}\leq\frac{1}{12}.

Combining (144), (148), and (149), we obtain (137). ∎

Finally, the result in Proposition 4.11 will let us to the desired differential inequality for the dissipation term and the proof of the main result Theorem 4.1.

Lemma 4.13.

Let $f$ be a solution of (98), and let $\bm{u}$ be given as in (98). Then for any $\gamma>0$ , there exist a sufficiently large positive constant $\Cr{const:D_{M}in}\geq 1$ and sufficiently small positive constants $\Cr{const:grad_{P}i},\Cr{const:Pi_{T}ime}>0$ which depend only on $n$ , $\|\phi\|_{L^{\infty}(\Omega)}$ , $\|\nabla\phi\|_{L^{\infty}(\Omega)}$ , $\lambda$ appeared in (20)(the lower bound of the Hessian of $\phi$ ), $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ defined in (14)(the bounds of the initial datum $f_{0}$ ), $\Cr{const:Pi_{M}in}$ , $\Cr{const:Pi_{M}ax}$ defined in (16)(the bounds of $\pi$ ), $\Cr{const:grad_{D}}$ defined in (19)(the bound of the gradient of $D$ ), $\Cr{const:4.Sobolev}$ appeared in (104), and $\gamma$ such that if (15), (17), and (18) hold, then,

(150)

\frac{d^{2}}{dt^{2}}F[f](t)\geq\gamma\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx-\frac{1}{12\Cr{const:Pi_{M}in}^{3}}\left(\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx\right)^{3}.

Proof.

By Lemma 4.5,

(151)

\int_{\Omega}|\bm{u}|^{2}f\,dx\leq\Cr{const:4.Poincare}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx,

provided the relation (111): $\Cr{const:grad_{P}i}\leq\frac{\Cr{const:Pi_{M}in}}{2\Cr{const:4.Sobolev}}.$ With this assumption, by (15) and (112), we have that,

(152)

\begin{split}&-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx+\int_{\Omega}D(x)|\nabla\bm{u}|^{2}f\,dx\\ &\geq-2(\lambda+1)\int_{\Omega}|\bm{u}|^{2}f\,dx+\Cr{const:D_{M}in}\int_{\Omega}|\nabla\bm{u}|^{2}f\,dx\\ &\geq\left(-2(\lambda+1)+\frac{\Cr{const:D_{M}in}}{\Cr{const:4.Poincare}}\right)\int_{\Omega}|\bm{u}|^{2}f\,dx.\end{split}

Note that $\Cr{const:4.Poincare}$ depends only on $n$ , $\Cr{const:InitMin}$ , $\Cr{const:InitMax}$ , $\Cr{const:Pi_{M}in}$ , $\Cr{const:Pi_{M}ax}$ , $\Cr{const:grad_{D}}$ , and $\|\phi\|_{L^{\infty}(\Omega)}$ , but is independent of $\Cr{const:D_{M}in}$ . Thus, for $\gamma>0$ , take $\Cr{const:D_{M}in}\geq 1$ sufficiently large such that

(153)

-2(\lambda+1)+\frac{\Cr{const:D_{M}in}}{\Cr{const:4.Poincare}}\geq\gamma\Cr{const:Pi_{M}ax}.

Further, we take $\Cr{const:D_{M}in}\geq 1$ sufficiently large and sufficiently small $\Cr{const:grad_{P}i},\Cr{const:Pi_{T}ime}>0$ such that the assumptions (138), (139), (140), (141), and (111) hold. Note that from Corollary 1.8 and Lemma 4.5, $\Cr{const:log_{f}}$ and $\Cr{const:4.Poincare}$ are independent of $\Cr{const:D_{M}in}$ , namely $\log f$ is bounded uniformly with respect to $\Cr{const:D_{M}in}$ . Then, we can use (137) and

(154)

\frac{d^{2}}{dt^{2}}F[f](t)\geq\gamma\Cr{const:Pi_{M}ax}\int_{\Omega}|\bm{u}|^{2}f\,dx-\frac{1}{12}\left(\int_{\Omega}|\bm{u}|^{2}f\,dx\right)^{3}.

Finally, using the bounds of $\pi$ in (16), we obtain (150). ∎

Now we are ready to prove the main result of this Section.

Proof of Theorem 4.1.

For any $\gamma>0$ , using Lemma 4.13, take sufficiently large positive number $\Cr{const:D_{M}in}\geq 1$ and sufficiently small positive numbers $\Cr{const:Pi_{T}ime},\Cr{const:grad_{P}i}>0$ . Then define,

g(t)=-\frac{d}{dt}F[f](t)=\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx,

and $c=\gamma,\quad p=3,$ together with $d=\frac{1}{12\Cr{const:Pi_{M}in}^{3}}.$ Thus, if

(155)

g(0)=\int_{\Omega}\pi(x,0)|\nabla(D(x)\log f_{0}(x)+\phi(x))|^{2}f_{0}\,dx<\left(12\gamma\Cr{const:Pi_{M}in}^{3}\right)^{\frac{1}{2}},

then by Lemma 3.12

(156)

\begin{split}g(t)&=\int_{\Omega}\pi(x,t)|\bm{u}|^{2}f\,dx\\ &\leq\left(\left(\int_{\Omega}\pi(x,0)|\nabla(D(x)\log f_{0}(x)+\phi(x))|^{2}f_{0}\,dx\right)^{-2}-\frac{1}{12\gamma\Cr{const:Pi_{M}in}^{3}}\right)^{-\frac{1}{2}}e^{-\gamma t}.\end{split}

Therefore, by taking $\Cr{const:4.Initial_{E}nergy}=(12\gamma\Cr{const:Pi_{M}in}^{3})^{\frac{1}{2}}$ and

(157)

\Cr{const:4.Exponential_{C}oefficient}=\left(\left(\int_{\Omega}\pi(x,0)|\nabla(D(x)\log f_{0}(x)+\phi(x))|^{2}f_{0}\,dx\right)^{-2}-\frac{1}{12\gamma\Cr{const:Pi_{M}in}^{3}}\right)^{-\frac{1}{2}},

we have finished the proof of Theorem 4.1. ∎

5. Conclusion

In this work, we considered generalized nonlinear Fokker-Planck type equations with inhomogeneous diffusion and with variable mobility parameters. Such systems appear as a part of grain growth modeling in polycrystalline materials. Using new reinterpretation of the classical entropy method under settings of the bounded domain with periodic boundary conditions and non-convexity assumptions on the potential function, we obtained the long time behavior of the solutions.

Acknowledgments

The work of Yekaterina Epshteyn was partially supported by NSF DMS-1905463 and by NSF DMS-2118172. The work of Chun Liu was partially supported by NSF DMS-1950868 and by NSF DMS-2118181. The work of Masashi Mizuno was partially supported by JSPS KAKENHI Grant Numbers JP22K03376 and JP23H00085.

References

[1] Anton Arnold, Peter Markowich, Giuseppe Toscani, and Andreas Unterreiter. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Partial Differential Equations, 26(1-2):43–100, 2001.
[2] Ralph Baierlein. Thermal Physics. Cambridge University Press, 1999.
[3] Patrick Bardsley, Katayun Barmak, Eva Eggeling, Yekaterina Epshteyn, David Kinderlehrer, and Shlomo Ta’asan. Towards a gradient flow for microstructure. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28(4):777–805, 2017.
[4] K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp, and S. Ta’asan. Critical events, entropy, and the grain boundary character distribution. Phys. Rev. B, 83:134117, Apr 2011.
[5] K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, and S. Ta’asan. Geometric growth and character development in large metastable networks. Rend. Mat. Appl. (7), 29(1):65–81, 2009.
[6] Katayun Barmak, Anastasia Dunca, Yekaterina Epshteyn, Chun Liu, and Masashi Mizuno. Grain growth and the effect of different time scales. In Research in mathematics of materials science, volume 31 of Assoc. Women Math. Ser., pages 33–58. Springer, Cham, [2022] ©2022.
[7] Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, and Shlomo Ta’asan. An entropy based theory of the grain boundary character distribution. Discrete Contin. Dyn. Syst., 30(2):427–454, 2011.
[8] Batuhan Bayir, Yekaterina Epshteyn, and William M Feldman. Global well-posedness of a nonlinear Fokker-Planck type model of grain growth. arXiv:2502.13151, 2025.
[9] R. Stephen Berry, Stuart Alan Rice, and John Ross. Physical chemistry. Topics in physical chemistry series. Oxford University Press, 2nd ed. edition, 2000.
[10] Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
[11] José A. Cañizo, José A. Carrillo, Philippe Laurençot, and Jesús Rosado. The Fokker-Planck equation for bosons in 2D: well-posedness and asymptotic behavior. Nonlinear Anal., 137:291–305, 2016.
[12] J. A. Carrillo and G. Toscani. Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations. Math. Methods Appl. Sci., 21(13):1269–1286, 1998.
[13] José A. Carrillo, María D. M. González, Maria P. Gualdani, and Maria E. Schonbek. Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience. Comm. Partial Differential Equations, 38(3):385–409, 2013.
[14] Pierre Degond, Maxime Herda, and Sepideh Mirrahimi. A Fokker-Planck approach to the study of robustness in gene expression. Math. Biosci. Eng., 17(6):6459–6486, 2020.
[15] Weinan E, Tiejun Li, and Eric Vanden-Eijnden. Applied stochastic analysis, volume 199 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2019.
[16] Yekaterina Epshteyn, Chang Liu, Chun Liu, and Masashi Mizuno. Nonlinear inhomogeneous Fokker-Planck models: energetic-variational structures and long-time behavior. Anal. Appl. (Singap.), 20(6):1295–1356, 2022.
[17] Yekaterina Epshteyn, Chang Liu, Chun Liu, and Masashi Mizuno. Local well-posedness of a nonlinear Fokker-Planck model. Nonlinearity, 36(3):1890, feb 2023.
[18] Yekaterina Epshteyn, Chun Liu, and Masashi Mizuno. Large time asymptotic behavior of grain boundaries motion with dynamic lattice misorientations and with triple junctions drag. Communications in Mathematical Sciences, 19(5):1403–1428, 2021.
[19] Yekaterina Epshteyn, Chun Liu, and Masashi Mizuno. Motion of Grain Boundaries with Dynamic Lattice Misorientations and with Triple Junctions Drag. SIAM J. Math. Anal., 53(3):3072–3097, 2021.
[20] Yekaterina Epshteyn, Chun Liu, and Masashi Mizuno. A stochastic model of grain boundary dynamics: a Fokker-Planck perspective. Math. Models Methods Appl. Sci., 32(11):2189–2236, 2022.
[21] J. L. Ericksen. Introduction to the thermodynamics of solids, volume 131 of Applied Mathematical Sciences. Springer-Verlag, New York, revised edition, 1998.
[22] Lawrence C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
[23] C. W. Gardiner. Handbook of stochastic methods for physics, chemistry and the natural sciences, volume 13 of Springer Series in Synergetics. Springer-Verlag, Berlin, third edition, 2004.
[24] Mi-Ho Giga, Arkadz Kirshtein, and Chun Liu. Variational modeling and complex fluids. In Handbook of mathematical analysis in mechanics of viscous fluids, pages 73–113. Springer, Cham, 2018.
[25] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
[26] Timothy Gowers, June Barrow-Green, and Imre Leader, editors. The Princeton companion to mathematics. Princeton University Press, Princeton, NJ, 2008.
[27] Jingwei Hu, Jian-Guo Liu, Yantong Xie, and Zhennan Zhou. A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration. J. Comput. Phys., 433:Paper No. 110195, 23, 2021.
[28] Ansgar Jüngel. Entropy methods for diffusive partial differential equations. SpringerBriefs in Mathematics. Springer, [Cham], 2016.
[29] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1967.
[30] Gary M. Lieberman. Second order parabolic differential equations. World Scientific Publishing Co. Inc., River Edge, NJ, 1996.
[31] Fang-Hua Lin, Chun Liu, and Ping Zhang. On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math., 58(11):1437–1471, 2005.
[32] P. A. Markowich and C. Villani. On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis. In VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999), volume 19, pages 1–29. Sociedade Brasileira de Matemática, Rio de Janeiro, 2000.
[33] Donald Allan McQuarrie. Statistical mechanics. Harper’s chemistry series. Harper & Row, 1976.
[34] Grigorios A. Pavliotis. Stochastic processes and applications, volume 60 of Texts in Applied Mathematics. Springer, New York, 2014. Diffusion processes, the Fokker-Planck and Langevin equations.
[35] H. Risken. The Fokker-Planck equation, volume 18 of Springer Series in Synergetics. Springer-Verlag, Berlin, second edition, 1989. Methods of solution and applications.
[36] R. Tyrrell Rockafellar. Convex analysis, volume No. 28 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1970.
[37] Silvio R. A. Salinas. Introduction to statistical physics. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 2001. Translated from the Portuguese.
[38] Gabriele Sicuro, Peter Rapčan, and Constantino Tsallis. Nonlinear inhomogeneous fokker-planck equations: Entropy and free-energy time evolution. Phys. Rev. E, 94:062117, Dec 2016.
[39] Yiwei Wang and Chun Liu. Some recent advances in energetic variational approaches. Entropy, 24(5):Paper No. 721, 26, 2022.
[40] Hao Wu, Xiang Xu, and Chun Liu. On the general Ericksen-Leslie system: Parodi’s relation, well-posedness and stability. Arch. Ration. Mech. Anal., 208(1):59–107, 2013.