Generalized Solutions to Nonlinear Fokker–Planck Equations with Linear Drift

Abstract

Existence and long-time behaviour of solutions to nonlinear Fokker–Planck equations (NFPEs) with linear drift are studied.

Dedicated to Michael Röckner at his 60th birthday.

1 The Problem

Here, we shall consider the equation

$$\begin{aligned} \begin{array}{l} u_t(t,x)+\mathrm {div}_x(D(x)u(t,x))-{\Delta }_x{\beta }(u(t,x))=0 \text{ in } (0,{\infty })\times {\mathbb {R}}^d, \\ u(0,x)=u_0(x),\ \ x\in {\mathbb {R}}^d,\ d\ge 1, \end{array} \end{aligned}$$

(1.1)

where

• (i) $D\in L^{\infty }({\mathbb {R}}^d;{\mathbb {R}}^d)$, $\mathrm {div}\ D\in L^{\infty }({\mathbb {R}}^d)$.

  ${\beta }$ is continuous, monotonically nondecreasing, ${\beta }(0)=0$,

  $|{\beta }(r)|\le C_{1}|r|^{m}+C_{2},$ $\forall r\in {\mathbb {R}}$, where $1\le m<{\infty }$.

Equation (1.1) describes the evolution of a probability density $u={P}$ associated to the Markovian stochastic processes with drift coefficients $(D_{i})_{i=1}^{d}=D$ and diffusion $\sigma _{ij}=\delta _{ij}.$ Moreover, it is related to the anomalous diffusion which describes particle transport in irregular media. In the special case $D\equiv 0$, (1.1) reduces to the nonlinear porous media equation in ${\mathbb {R}}^d$.

In 1D, Eq. (1.1) can be derived from the entropy functional

$$\begin{aligned} S[u]=\int _{{\mathbb {R}}}\Phi [u(x)]dx, \end{aligned}$$

(1.2)

where

$$\begin{aligned} \Phi \in C^{{\infty }}(0,{\infty }),\ \lim \limits _{r\rightarrow 0}\Phi ^{\prime }(r)={\infty } \text{ and } \Phi ^{\prime \prime }(r)<0 \text{ for } r>0\text{. } \end{aligned}$$

(1.3)

The corresponding Fokker–Planck equation is

$$\begin{aligned} P_t+\left( H(x)P-\frac{1}{\alpha }\,(\Phi (P)-P \Phi '(P))_x\right) _x=0. \end{aligned}$$

(1.4)

Here the drift function H is the gradient of a potential V (i.e., $H=-\frac{dV}{dx}$) and the constant ${\alpha }$ represents the strength of fluctuations [5]. A similar approach applies to higher dimensions.

In the special case of the Boltzmann–Gibbs entropy

$$\begin{aligned} S[u]=-\int u(x)\log u(x)dx, \end{aligned}$$

Equation (1.4) reduces to

$$\begin{aligned} P_{t}+P_{x}-\frac{1}{{\alpha }}\,P_{xx}=0, \end{aligned}$$

(1.5)

while, for the entropy functional

$$\begin{aligned} S[u]=\frac{1}{p-1}\,\int (|u|^{p}-u)dx,\ p>1, \end{aligned}$$

(1.6)

Equation (1.4) with $H\equiv 1$ reads as the Plastino and Plastino model [7]

$$\begin{aligned} P_{t}+P_{x}-\frac{1}{{\alpha }}\,((P)^{p})_{xx}=0. \end{aligned}$$

Assumption (i) agrees with the key entropy condition (1.3). Indeed, if $\Phi \in C^1(0,{\infty })\cap C[0,{\infty })$ is a solution to the equation

$$\begin{aligned} \Phi (r)-r\Phi '(r)=\beta (r),\ \forall r>0;\ \Phi '(0)={\infty }, \end{aligned}$$

(1.7)

such that

$$\Phi ''(r)<0,\ \Phi '(r)\ge 0,\ \forall r\in \mathbb {R},$$

where ${\beta }$ satisfies (i), the NFPE reduces to (1.1) such that (1.3) holds. We note that, in particular, assumption (i) is satisfied for ${\beta }(u)=\frac{1}{{\alpha }}\ln (1+u)$

, that is, for the Fokker–Planck equation of classical bosons (see [4, 5])

$$\begin{aligned} P_{t}+(DP)_{x}+\frac{1}{{\alpha }}\,(\ln (1+P))_{xx}=0. \end{aligned}$$

In [1], Eq. (1.1), was studied the existence of an entropy solution for the Fokker–Planck equation

$$\begin{aligned} \begin{array}{ll} u_t+\mathrm{div}(D(x,u)u)-{\Delta }\beta (u)=0,&{} \text{ in } (0,T)\times {\mathbb {R}}^d,\\ u(0,x)=u_0(x),\end{array} \end{aligned}$$

(1.8)

where $D(x,u)\equiv b(u)$, with b continuous. In this work, we shall confine to the case of linear drift $D(x,u)\equiv D(x).$

2 The Existence and Uniqueness of a Generalized Solution

To (1.1) we associate the operator $A:L^{1}({\mathbb {R}}^{d})\rightarrow L^{1}({\mathbb {R}}^{d})$ defined as the closure $\overline{A}_{1}$ in $L^{1}(\mathbb {R}^{d})\times L^{1}(\mathbb {R}^{d})$ of the operator

$$\begin{aligned} \begin{array}{rcl} A_{1}u &{} {=} &{} -{\Delta }{\beta }(u)+\mathrm {div}(D(x)u),\ \forall u\in D(A_{1}), \\ D(A_{1}) &{} {=} &{} \{u\in L^{1}({\mathbb {R}}^{d})\cap L^{\infty }({\mathbb {R}}^{d}),\ {\beta }(u)\in H^{1}({\mathbb {R}}^{d}),\ A_{1}u\in L^{1}(\mathbb {R}^{d})\}. \end{array} \end{aligned}$$

(2.1)

We have also

Lemma 2.1

The operator $A_{1}$ is accretive in $L^{1}({\mathbb {R}}^{d})$ and

$$\begin{aligned} L^{1}(\mathbb {R}^{d})\cap L^{\infty }(\mathbb {R}^{d})\subset & {} R(I+{\lambda } A_{1}),\ \forall {\lambda }>0. \end{aligned}$$

(2.2)

$$\begin{aligned} (I+{\lambda }A_{1})^{-1}f\ge & {} 0\text { in }\mathbb {R}^{d}\text { if }f\ge 0\ \ \text {in }\mathbb {R}^{d} \end{aligned}$$

(2.3)

$$\begin{aligned} \int _{\mathbb {R}^{d}}(I+{\lambda }A_{1})^{-1}f\,dx= & {} \int _{\mathbb {R} ^{d}}f\,dx\ \ \text {in }\mathbb {R}^{d}. \end{aligned}$$

(2.4)

Proof

The accretivity of $A_{1}$ follows by multiplying the equation

$$\begin{aligned} u-\bar{u}+{\lambda }(A_{1}u-A_{1}\bar{u})=f-\bar{f},\ \ u,\bar{u}\in D(A_{1}), \end{aligned}$$

in the duality pair $_{H^{-1}(\mathbb {R}^{d})}\langle \cdot ,\cdot \rangle _{H^{1}(\mathbb {R}^{d})}$ with ${\mathcal {X}}_{\varepsilon }(u-\bar{u})$ and integrate over $\mathbb {R}^{d}$, where ${\mathcal {X}}_{\varepsilon }$ is a smooth approximation of the sign function for ${\varepsilon }\rightarrow 0$.

More precisely, $\chi _{\varepsilon }$ is defined by

$$\chi _{\varepsilon }(r)=\left\{ \begin{array}{rll} -1&{} \text{ for } &{}r<-{\varepsilon },\\ \frac{1}{{\varepsilon }}\,r&{} \text{ for } &{}|r|<{\varepsilon },\\ 1&{} \text{ for } &{}r>{\varepsilon }.\end{array}\right. $$

To prove (2.2), we fix $f\in L^{1}(\mathbb {R}^{d})\cap L^{\infty }(\mathbb {R}^{d})\subset L^{2}(\mathbb {R}^{d})$ and consider the equation $u+{\lambda }A_{1}u=f$, that is,

$$\begin{aligned} u-{\lambda }{\Delta }{\beta }(u)+{\lambda }\,\mathrm {div}(Du)=f\text { in }{\mathcal {D}}^{\prime }(\mathbb {R}^{d}),\ {\lambda }>0. \end{aligned}$$

(2.5)

(Here, ${\mathcal {D}}'({\mathbb {R}}^d)$ is the space of distributions on ${\mathbb {R}}^d$.)

We set ${\beta }_{\varepsilon }=\frac{1}{\varepsilon }\,{\beta }(I+{\varepsilon }{\beta })^{-1})$ and approximate (2.5) by

$$\begin{aligned} u-{\lambda }{\Delta }({\beta }_{\varepsilon }(u)+{\varepsilon }u)+{\lambda } \,\mathrm {div}(Du)=f\text { in }\mathbb {R}^{d}, \end{aligned}$$

(2.6)

Equivalently,

$$\begin{aligned} ({\varepsilon }+{\beta }_{\varepsilon })^{-1}(v)-{\lambda }{\Delta }v+{ \lambda }\,\mathrm {div}(D({\varepsilon }+{\beta }_{\varepsilon })^{-1}(v))=f \text { in }\mathbb {R}^{d}. \end{aligned}$$

(2.7)

The operator $v\rightarrow {\beta }_{\varepsilon }^{-1}(v)-{\lambda }{\Delta }v$ is coercive and maximal monotone in $H^{1}(\mathbb {R}^{d})\times H^{-1}(\mathbb {R}^{d})$ and so, for each $w\in L^{2}(\mathbb {R}^{d})$,

$$\begin{aligned} ({\varepsilon }+{\beta }_{\varepsilon })^{-1}(v)-{\lambda }{\Delta }v=-{\lambda }\,\mathrm {div}(D({\varepsilon }+{\beta }_{\varepsilon })^{-1}(w))+f\ \ \text { in }\mathbb {R}^{d} \end{aligned}$$

has a unique solution $v=F(w)\in H^{1}(\mathbb {R}^{d})$.

By the contraction principle, for ${\lambda }>\frac{1}{2L}\,\Vert D\Vert _{\infty }$, Eq. (2.7) has a unique solution $v_{\varepsilon }\in H^1({\mathbb { R}}^d)$. This extends to all ${\lambda }>0$.

We have by (2.6)

$$\begin{array}{c} |u_{\varepsilon }|_{p}\le |f|_{p},\ \forall {\varepsilon }>0,\ p\in [1,{\infty }), \\ |\nabla {\beta }_{\varepsilon }(u_{\varepsilon })|_{2}^{2}+{\varepsilon } |\nabla u_{\varepsilon }|_{2}^{2}\le C,\ \forall {\varepsilon }>0. \end{array}$$

(Here, $|\cdot |_p$, $1\le p\le {\infty }$, is the norm of $L^p({\mathbb {R}}^d)$.)

On a subsequence ${\varepsilon }\rightarrow 0$, we have

$$\begin{aligned} \begin{array}{rcll} u_{\varepsilon } &{} \rightarrow &{} u &{} \text{ weakly } \text{ in } L^{p},\ p\in (1,{ \infty }), \\ {\beta }_{\varepsilon }(u_{\varepsilon }) &{} \rightarrow &{} \eta &{} \text{ weakly } \text{ in } H^{1}, \\ {\Delta }({\beta }_{\varepsilon }(u_{\varepsilon })+{\varepsilon } u_{\varepsilon }) &{} \rightarrow &{} {\Delta }\eta &{} \text{ weakly } \text{ in } H^{-1}, \\ \mathrm {div}(Du_{\varepsilon }) &{} \rightarrow &{} \mathrm {div}(Du) &{} \text{ in } H^{-1}. \end{array} \end{aligned}$$

$$\begin{aligned} \limsup _{{\varepsilon }\rightarrow 0}\int _{{\mathbb {R}}^{d}}{\beta }_{\varepsilon }(u_{\varepsilon })u_{\varepsilon }\,dx\le -{\lambda }\int _{{\mathbb {R}} ^{d}}|\nabla \eta |^{2}dx-\int _{{\mathbb {R}}^{d}}fu\,dx=\int _{{\mathbb {R}}^d}\eta u\,dx, \end{aligned}$$

$$\begin{aligned} u-{\lambda }{\Delta }\eta +{\lambda }\,\mathrm {div}(Du)=f \text{ in } { \mathcal {D}}^{\prime }({\mathbb {R}}^{d}). \end{aligned}$$

Hence $\eta ={\beta }(u),$ a.e. in ${\mathbb {R}}^{d}$ and $u+{\lambda } A_{1}u=f$, as claimed.

Proposition 2.2

Under assumption (i), the operator $A=\overline{A} _{1}$ is m-accretive in $L^{1}({\mathbb {R}}^{d})$. Moreover, one has for all ${\lambda }>0$

$$\begin{aligned} (I+{\lambda }A)^{-1}= & {} (I+{\lambda }A_{1})^{-1} \text{ on } L^{\infty }({ \mathbb {R}}^{d})\cap L^{1}({\mathbb {R}}^{d}) \end{aligned}$$

(2.8)

$$\begin{aligned} (I+{\lambda }A)^{-1}f\ge & {} 0 \text{ if } f\ge 0 \text{ in } {\mathbb {R}} ^{d} \end{aligned}$$

(2.9)

$$\begin{aligned} \int _{{\mathbb {R}}^{d}}(I+{\lambda }A)^{-1}f\,dx= & {} \int _{{\mathbb {R}} ^{d}}f\,dx,\ \ \forall {\lambda }>0. \end{aligned}$$

(2.10)

It follows also that $\overline{D(A)}=L^{1}({\mathbb {R}}^{d}).$

By Proposition 2.2, the finite difference scheme

$$\begin{aligned} \begin{array}{c} u_{h}(t)+hAu_{h}(t)=u_{h}(t-h),\ h>0,\ t\ge 0, \\ \ \ u_{h}(t)=u_{0},\ \ \text{ for } t\le 0, \end{array} \end{aligned}$$

(2.11)

has a unique solution $u_{h}$ and, by then, by the Crandall and Liggett exponential formula (see [3]),

$$\begin{aligned} u_{h}(t)\rightarrow u(t) \text{ strongly } \text{ in } L^{1}({\mathbb {R}}^{d}), \end{aligned}$$

(2.12)

uniformly on compact intervals.

The function u is called the mild solution to Eq. (1.1).

Now, we can formulate the main existence result.

Theorem 2.3

Under assumptions (i), for each $u_{0}\in L^{1}({ \mathbb {R}}^{d})\cap L^{\infty }({\mathbb {R}}^{d})$, Eq. (1.1) has a unique mild solution $u\in C([0,T];L^{1}({\mathbb {R}}^{d}))$, $\forall T>0$. Moreover, $S(t)u_{0}=u(t)$ is a continuous semigroup of contractions in $L^{1}({\mathbb {R}}^{d})$,

$$\begin{aligned}&|S(t)u_{0}|_{p}\le |u_{0}|_{p},\ \ \forall u_{0}\in L^{p}({\mathbb {R}} ^{d}),\ 1\le p\le {\infty } \end{aligned}$$

(2.13)

$$\begin{aligned}&u(t,x)\ge 0,\ \text{ a.e. } x\in {\mathbb {R}}^{d} \text{ if } u_{0}(x)\ge 0, \text{ a.e. } x\in {\mathbb {R}}^{d}, \end{aligned}$$

(2.14)

$$\begin{aligned}&\displaystyle \int _{{\mathbb {R}}^{d}}u(t,x)dx=\int _{{\mathbb {R}} ^{d}}u_{0}(x)dx,\ \forall t\ge 0. \end{aligned}$$

(2.15)

Definition 2.4

The function $u\in C([0,T];L^{1}({\mathbb {R}}^{d}))$ is said to be a generalized solution to (1.1) if

$$\begin{aligned} \begin{array}{ll} \displaystyle \frac{{\partial }u}{{\partial }t}+\mathrm {div}_{x}(D(x)u)-{ \Delta }_{x}{\beta }(u)=0 &{} \text{ in } {\mathcal {D}}^{\prime }((0,T)\times { \mathbb {R}}^{d}), \\ u(0,x)=u_{0}(x) &{} \text{ in } {\mathbb {R}}^{d}. \end{array} \end{aligned}$$

(2.16)

By (2.11)–(2.12), we see that

Theorem 2.5

Under assumption (i), for each $u_{0}\in L^{1}({ \mathbb {R}}^{d})\cap L^{\infty }({\mathbb {R}}^{d})$, the mild solution u to (2.1) is a generalized solution. Moreover, there is at most one generalized solution $u\in C([0,T];L^{1}({\mathbb {R}}^{d})\cap L^{\infty }((0,T)\times {\mathbb {R}}^{d}).$

The uniqueness part of Theorem 2.5 follows as in [1] and it will be omitted.

3 Long-Time Dynamical Behavior

Let S(t) be the semigroup of contractions generated on $\overline{D(A)}$ under assumption (i).

Definition 3.1

A real valued function $\psi :L^{1}({\mathbb {R}} ^{d})\rightarrow {\mathbb {R}}^{+}$ is a Lyapunov function for S(t) if $\psi (S(t)u_{0})\le \psi (u_{0}),\ \forall t\ge 0,\ \forall u_{0}\in D(\psi )\cap \overline{D(A)},$ where $D(\psi )=\{u_{0}\in L^{1}({\mathbb {R}} ^{d});\ \psi (u_{0})<{\infty }\}.$

As we shall see later on, the free energy of the system (the so-called H-functional) is the best candidate for the Lyapunov functions.

Again, by (2.11) and (2.12), we see that

Proposition 3.2

Assume that $\psi :L^{1}(\mathbb {R}^{d})\rightarrow \mathbb {R}^{+}$ is lower semicontinuous and, for all $\lambda >0,$

$$\begin{aligned} \psi ((I+{\lambda }A)^{-1}u_{0})\le \psi (u_{0}),\ \forall u_{0}\in D(\psi )\cap \overline{D(A)}. \end{aligned}$$

(3.1)

Then, $\psi$ is a Lyapunov function for the semigroup S(t).

Here, $D(\psi )=\{u;\ \psi (u)<{\infty }\}.$

We shall look at Lyapunov functions of the form

$$\begin{aligned} \psi (u)=\int _{{\mathbb {R}}^{d}}j(u(x))dx,\ \ \forall u\in L^{1}({\mathbb {R}} ^{d}), \end{aligned}$$

(3.2)

where $j:{\mathbb {R}}\rightarrow {\mathbb {R}}^{+}$ is convex, lower semicontinuous and $j(0)=0$. Then, as well known, $\psi :L^{1}({\mathbb {R}} ^{d})\rightarrow {\mathbb {R}}$ is convex, lower semicontinuous and

$$\begin{aligned} D(\psi )=\{u\in L^{1}({\mathbb {R}}^{d});\ j(u)\in L^{1}({\mathbb {R}}^{d})\}. \end{aligned}$$

We have (see [1]).

Theorem 3.3

Under hypothesis (i), the function $\psi$ is a Lyapunov function for S(t).

Remark 3.4

In particular, it follows by Proposition 3.2 that the operator A is m-completely accretive in sense of [2].

4 The H-Theorem

The so-called H-theorem amounts to saying that, for $t\rightarrow {\infty }$, $S(t)u_0\rightarrow v$, where v is a stationary solution to Eq. (1.1), that is, $Av=0$ (see [5, 6]).

Since, as easily seen by (2.11), for each $\ell$ and $u_{0}\in \overline{D(A)}$,

$$\begin{aligned} \int _{{\mathbb {R}}^{d}}|(S(t)u_{0})(x+\ell )-(S(t)u_{0})(x)|dx\le \int _{{\mathbb {R}}^d}|u_{0}(x+\ell )-u_0(x)|dx, \end{aligned}$$

by the Kolmogorov compactness theorem, the trajectory $\{S(t)u_{0};\ t\ge 0\}$ is compact in $L_{loc}^{1}(R^{d})$ and in every $L_{loc}^{p}(R^{d})$, $1\le p<\infty$, if $u_{0}\in L^{1}(R^{d})\cap L^{\infty }(R^{d})$.

Hence the $\omega$-limit set

$$\omega (u_{0})=\left\{ v=\lim \limits _{t_n\rightarrow {\infty }}S(t_{n})u_{0} \text{ in } L_{loc}^{1}({ \mathbb {R}}^{d})\right\} $$

is nonempty and, if $u_{0}\in L^{1}(R^{d})\cap L^{\infty }(R^{d})$, we have also by (2.13) that

$$\begin{aligned} {\omega }(u_{0})=\left\{ v=\underset{t_{n}\rightarrow \infty }{\mathrm {weak\,\, limit}} \ \ S(t_{n})u_{0} \text{ in } \text{ each } L^{p}({\mathbb {R}} ^{d})\right\} . \end{aligned}$$

By weak lower-semicontinuity of $\psi$, we have

$$ \psi (v)\le \lim _{t\rightarrow {\infty }}\psi (S(t)u_{0}),\ \ \forall v\in {\omega }(u_{0}). $$

If $\psi$ is continuous on $L^{1}$ and $\{S(t)u_{0},\ t\ge 0\}$ is compact, then we have

$$\begin{aligned} \psi (S(t)v)\equiv \psi (v),\ \forall t\ge 0,\ v\in {\omega }(u_{0}), \end{aligned}$$

(4.1)

which is a weak form of the H-theorem.

To give a specific example, we assume that

$$\begin{aligned}&D=-\nabla V,\ V\in C^1({\mathbb {R}}^d),\ V\ge 0,\end{aligned}$$

(4.2)

$$\begin{aligned}&\beta \in C^1({\mathbb {R}}),\ \beta (0)=0,\ \beta '(r)>0,\ \forall r>0,\end{aligned}$$

(4.3)

$$\begin{aligned}&\displaystyle \inf \{\Vert D(x)\Vert ;\,x\in {\mathbb {R}}^d\}>0. \end{aligned}$$

(4.4)

Then, as easily seen by (4.2), (4.3), the energy functional

$$ E(u)=\int _{{\mathbb {R}}^d}(V(x)u(x)-\Phi (u(x)))dx,$$

where $\Phi$ is given by (1.7), and so

$$\Phi ''(r)=-\frac{\beta '(r)}{r}\,\ \forall r>0.$$

We note that E is convex and is a Lyapunov functional for the semigroup S(t). Indeed, taking into account that ${\partial }E(u)=V-\Phi '(u)$, we get that

$$\begin{aligned} \begin{array}{lcl} \langle {\partial }E(u),A_1u\rangle &{}=&{} \displaystyle \int _{{\mathbb {R}}^d}(V-\Phi '(u))(-\Delta \beta (u)+\mathrm{div}(D u))dx\\ &{}=&{}\displaystyle \int _{{\mathbb {R}}^d}\left( \frac{(\beta '(u))^2}{u}\,|\nabla u|^2+u|\nabla V|^2\right) dx\ge 0,\\ &{}&{}\qquad \qquad \qquad \qquad \qquad \;\; \forall u\in D(A_1),\ u\ge 0,\end{array}\end{aligned}$$

(4.5)

and, by density, this implies that

$$\begin{aligned} E(S(t)u_0)\le {\mathbb {E}}(u_0),\ \forall t\ge 0.\end{aligned}$$

(4.6)

Moreover, by (4.4), (4.6) and (2.11), we see that

$$\begin{array}{ll} E(u_h(ih))-E((u_h(i-1)h)) \\ \qquad \le -h\displaystyle \int _{{\mathbb {R}}^d}\left( \frac{(\beta '(u_h(ih)))^2}{u_h(ih)}\right) \,|\nabla u_h(ih)|^2+u_h(ih)|\nabla V|^2dx\\ \qquad \le -\rho \displaystyle \int _{{\mathbb {R}}^d}u_h(ih)dx,\ \forall i=1,2,\ldots ,h>0,\end{array}$$

where $\rho >0.$ This yields

$$\frac{1}{t-s}\,(E(S(t)u_0)-E(S(s)u_0))\le \rho |S(t)u_0|_{1},\ \forall t>s>0,$$

and, therefore,

$$E(S(t)u_0)\le \exp (-\rho t)|u_0|_{1},\ \forall t\ge 0.$$

Hence, if $u_{\infty }\in \omega (u_0),$ we have $E(u_{\infty })=0$. Assume further that

$$\begin{aligned} \inf _{x\in {\mathbb {R}}^d}V(x)>\sup _{r>0}\frac{\Phi (r)}{r}.\end{aligned}$$

(4.7)

Then the latter implies that $u_{\infty }=0.$ We have, therefore,

Theorem 4.1

Under assumptions (4.2)–(4.6), we have

$$\lim _{t\rightarrow {\infty }}S(t)u_0=0 \text{ in } L^1({\mathbb {R}}^d) \text{ for } \text{ each } u_0\in L^1({\mathbb {R}}^d).$$

5 Final Remarks

For $D=-\nabla V$, we associate with Eq. (1.1) the free energy functional

$$\begin{aligned} F(u)=\int _{{\mathbb {R}}^d}\Phi (u)dx+\int _{{\mathbb {R}}^d}Vu\,dx,\end{aligned}$$

(5.1)

where $\Phi$ satisfies (1.7) and $V\in C^{\infty }({\mathbb {R}}^d)$ is such that

$$V(x)\ge 0,\ |\nabla V(x)|\le C(V(x)+1),\ \forall x\in {\mathbb {R}}^d.$$

On the set

$$K=\Big \{u:{\mathbb {R}}^d\rightarrow [0,{\infty }) \text{ measurable }, \ \int _{{\mathbb {R}}^d}u(x)dx=1,\ \int _{{\mathbb {R}}^d}|x|^2u(x)dx<{\infty }\Big \},$$

consider the iterative scheme

$$\begin{aligned} u_{k+1}=\mathrm{arg}\min _{u\in K}\left\{ \frac{1}{2h}\,d^2(u,u_k)+F(u)\right\} ,\end{aligned}$$

(5.2)

where d is the Wasserstein distance (see, e.g., [6]).

Consider the sequence $u^h:[0,T]\rightarrow {\mathbb {R}}^d$ of the step functions

$$\begin{aligned} u^h(t)=u_k \text{ for } t\in [kh,(k+1)h],\ k=0,1.\end{aligned}$$

(5.3)

Problem

Does the sequence $\{u^h\}$ strongly converge to $S(t)u_0$ for $h\rightarrow 0$?

The answer is positive (see [6]) if $\Phi (u)=u\log u$ (the case of Gibbs–Boltzmann entropy), that is, for the Fokker–Planck equation

$$u_t+\mathrm{div}(D u)-\Delta u=0 \text{ in } (0,T)\times {\mathbb {R}}^d$$

and one might suspect that it is true in this case for other functions $\Phi$ satisfying (1.7).