Hypocoercivity and sub-exponential local equilibria

Abstract

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincaré inequalities compared to the more classical weak Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.

Appendices

Weighted Poincaré inequalities

The goal of this appendix is to provide a proof of (10). Inequality (10) is not a standard weighted Poincaré inequality because the average in the right-hand side of the inequality involves the measure of the left-hand side, so that the right-hand side cannot be interpreted as a variance. Section A.1 is devoted to a reformulation of a spectral gap issue associated with Poincaré inequalities with weights into spectral considerations for a Schrödinger operator. We establish a criterion for Poincaré inequalities which is well adapted to the weights in (10). The average, however, corresponds to a standard variance. In Sect. A.2, we establish the result for the average which appears in (10).

1.1 Continuous spectrum and weighted Poincaré inequalities

Let us consider two probability measures on ${\mathbb R}^d$

$$\begin{aligned} \mathrm d\xi =e^{-\phi }\,\mathrm dv\quad \text{ and }\quad \mathrm d\nu =\psi \,\mathrm d\xi , \end{aligned}$$

where $\phi$ and $\psi >0$ are two measurable functions, and the weighted Poincaré inequality

$$\begin{aligned} \forall \,h\in \mathcal D({\mathbb R}^d),\quad \int _{{\mathbb R}^d}{|\nabla h|^2}\,\mathrm d\xi \ge \mathcal {C}_\star \int _{{\mathbb R}^d}{\left| h-\widehat{h}\right| ^2}\,\mathrm d\nu \end{aligned}$$

(24)

where $\widehat{h}=\int _{{\mathbb R}^d}{h}\,\mathrm d\nu$. The question we address here is: on which conditions on $\phi$ and $\psi$ do we know that (24) holds for some constant $\mathcal {C}_\star >0$ ? Our key example is

$$\begin{aligned} \phi (v)=\left\langle v\right\rangle ^\alpha +\log Z_\alpha \quad \text{ and }\quad \psi (v)=c_{\alpha ,\beta }^{-1}\,\left\langle v\right\rangle ^{-\beta } \end{aligned}$$

(25)

with $\alpha >0$, $\beta >0$, $Z_\alpha =\int _{{\mathbb R}^d}{e^{-\left\langle v\right\rangle ^\alpha }}\,\mathrm dv$ and $c_{\alpha ,\beta }=Z_\alpha ^{-1}\int _{{\mathbb R}^d}{\left\langle v\right\rangle ^{-\beta }e^{-\left\langle v\right\rangle ^\alpha }}\,\mathrm dv$.

Here we use a spectral property of Schrödinger type operators, which goes as follows. Let us consider a measurable function $\Phi$ on ${\mathbb R}^d$ such that

$$\begin{aligned} \sigma =\lim _{r\rightarrow +\infty }\inf _{w\in \mathcal D(B_r^c)\setminus \{0\}}\frac{\int _{{\mathbb R}^d}{\left( |\nabla w|^2+\Phi \,|w|^2\right) }\,\mathrm dv}{\int _{{\mathbb R}^d}{|w|^2}\,\mathrm dv}>0, \end{aligned}$$

where $B_r^c:=\left\{ v\in {\mathbb R}^d\,:\,|v|>r\right\}$ and $\mathcal D(B_r^c)$ denotes the space of smooth functions on ${\mathbb R}^d$ with compact support in $B_r^c$. According to Persson’s result [26, Theorem 2.1], the lower end $\sigma _\star$ of the continuous spectrum of the Schrödinger operator $-\,\Delta +\Phi$ is such that

$$\begin{aligned} \sigma _\star \ge \sigma \ge \lim _{r\rightarrow +\infty } \mathop {\mathrm {ess\,inf}}_{v\in B_r^c}\,\Phi (v)\,. \end{aligned}$$

If we replace $\int _{{\mathbb R}^d}{|w|^2}\,\mathrm dv$ by the weighted integral $\int _{{\mathbb R}^d}{|w|^2\,\psi }\,\mathrm dv$ for some measurable function $\psi$, we have the modified result that the operator $\mathcal L=\psi ^{-1}\left( -\,\Delta +\Phi \right)$ on $\mathrm L^2({\mathbb R}^d,\psi \,\mathrm dv)$, associated with the quadratic form

$$\begin{aligned} w\mapsto \int _{{\mathbb R}^d}{\left( |\nabla w|^2+\Phi \,|w|^2\right) }\,\mathrm dv \end{aligned}$$

has only discrete eigenvalues in the interval $(-\infty ,\sigma )$ where

$$\begin{aligned} \sigma =\lim _{r\rightarrow +\infty }\inf _{w\in \mathcal D(B_r^c)\setminus \{0\}}\frac{\int _{{\mathbb R}^d}{\left( |\nabla w|^2+\Phi \,|w|^2\right) }\,\mathrm dv}{\int _{{\mathbb R}^d}{|w|^2\,\psi }\,\mathrm dv}>0\,. \end{aligned}$$

To prove it, it is enough to observe that 0 is the lower end of the continuous spectrum of $\mathcal L-\sigma _\star \,\psi$, where $\sigma _\star$ is again defined as the lower end of the continuous spectrum of $\mathcal L$, and to apply [26, Theorem 2.1]. It is also straightforward to check that $\sigma _\star$ is such that

$$\begin{aligned} \sigma _\star \ge \sigma \ge \lim _{r\rightarrow +\infty }\mathsf q(r)=:\sigma _0\qquad \text{ where }\quad \mathsf q(r):=\mathop {\mathrm {ess\,inf}}_{B_r^c}\,\frac{\Phi }{\psi }\,. \end{aligned}$$

(26)

Note that $\sigma _0$ is either finite or infinite.

Relating the weighted Poincaré inequality (24) with the spectrum of $\mathcal L$ is then classical. With

$$\begin{aligned} \Phi =\tfrac{1}{4}\,|\nabla \phi |^2-\tfrac{1}{2}\,\Delta \phi , \end{aligned}$$

(27)

the spectral gap of $\mathcal L$ is equal to the optimal constant in the Poincaré inequality. Indeed, let $h=w\,e^{\phi /2}$ and observe that

$$\begin{aligned}&\int _{{\mathbb R}^d}{|\nabla h|^2}\,\mathrm d\xi =\int _{{\mathbb R}^d}{\left( |\nabla w|^2+\Phi \,|w|^2\right) }\,\mathrm dv,\\&\int _{{\mathbb R}^d}{\left| h-\widehat{h}\right| ^2}\,\mathrm d\nu =\int _{{\mathbb R}^d}{\left| w-\widetilde{w}\right| ^2\,\psi }\,\mathrm dv, \end{aligned}$$

where $\widetilde{w}=e^{-\phi /2}\int _{{\mathbb R}^d}{w\,\psi \,e^{-\phi /2}}\,\mathrm dv$ is the orthogonal projection of w, with respect to $\mathrm L^2(\psi \, \mathrm dv)$, onto the kernel of $\mathcal L$. The kernel of $\mathcal L$ is generated by the constant functions. With $\sigma _\star >0$, we know that the interval $(0,\sigma _\star )$ contains only eigenvalues, with finite dimensional eigenspaces, which may eventually accumulate, but only with $\sigma _\star$ as the adherence value. As a consequence, there is a lowest positive eigenvalue of $\mathcal L$, which is positive and determines the spectral gap.

Proposition 9

Let $\phi$ and $\psi >0$ be two measurable functions. Let $\Phi$ and $\sigma _0$ be defined respectively by (27) and  (26) and assume that $\sigma _0$ is nonnegative. Then inequality (24) holds for some positive, finite, optimal constant $\mathcal {C}_\star \ge \sigma _0$ if $\sigma _0$ is positive. Otherwise, if $\sigma _0=0$, then inequality (24) does not hold.

Proof

By construction, $\sigma$ is nonnegative and the infimum of the Rayleigh quotient

$$\begin{aligned} w\mapsto \frac{\int _{{\mathbb R}^d}{\left( |\nabla w|^2+\Phi \,|w|^2\right) }\,\mathrm dv}{\int _{{\mathbb R}^d}{|w|^2\,\psi }\,\mathrm dv} \end{aligned}$$

is achieved by $h\equiv \widehat{h}=1$, that is, by $w=\widetilde{w}=e^{-\phi /2}$, which moreover generates the kernel of $\mathcal L$. Hence we can interpret $\mathcal {C}_\star$ as the first positive eigenvalue, if there is any in the interval $(0,\sigma _\star )$, or $\mathcal {C}_\star =\sigma _\star$ if there is none. Notice that in the latter case $\sigma _\star$ is finite. If $\sigma _0=0$, it is easy to construct a sequence of functions which shows that inequality (24) holds only if $\mathcal {C}_\star =0$. $\square$

In the case of (25), we have $\frac{\Phi }{\psi } \sim \frac{1}{4}\,\alpha ^2\,c_{\alpha ,\beta } \left\langle v\right\rangle ^{2\,(\alpha -1)+\beta }$ as $|v|\rightarrow \infty$. Thus the condition $\beta \ge 2\,(1-\alpha )$ is necessary and sufficient for the inequality (24) to hold. The threshold case $\beta =2\,(1-\alpha )$ is remarkable: inequality (24) can be rewritten for any $\alpha \in (0,1)$ as the following weighted Poincaré inequality :

$$\begin{aligned} \forall \,h\in \mathcal D({\mathbb R}^d),\quad \int _{{\mathbb R}^d}{|\nabla h|^2\,e^{-\left\langle v\right\rangle ^\alpha }}\,\mathrm dv\ge \mathcal {C}_\star \int _{{\mathbb R}^d}{\frac{\left| h-\widehat{h}\right| ^2\,e^{-\left\langle v\right\rangle ^\alpha }}{\left( 1+|v|^2\right) ^{1-\alpha }}}\,\mathrm dv, \end{aligned}$$

(28)

for some constant $\mathcal {C}_\star \in (0,\sigma _0)$. The above computation shows that $\sigma _0=\alpha ^2/4$ and

$$\begin{aligned} \widehat{h}:=\frac{1}{z_\alpha }\int _{{\mathbb R}^d}{\frac{h\,e^{-\left\langle v\right\rangle ^\alpha }}{\left( 1+|v|^2\right) ^{1-\alpha }}}\,\mathrm dv,\quad z_\alpha =\int _{{\mathbb R}^d}{\frac{e^{-\left\langle v\right\rangle ^\alpha }}{\left( 1+|v|^2\right) ^{1-\alpha }}}\,\mathrm dv\,. \end{aligned}$$

Notice that (28) differs from (10), as the average in the right-hand side is not taken with respect to the same measure in both inequalities. The purpose of the next subsection is to deduce (10) from (28).

1.2 A weighted Poincaré inequality with a non-classical average

Corollary 10

Let the assumptions of Proposition 9 hold with $\sigma _0>0$ and let, additionally, $\psi$ be bounded, $\psi ^{-1}\in \mathrm L^1(\mathrm d\xi )$ and such that $\lim _{R\rightarrow +\infty }R^2\inf _{B_{2R}}\psi =+\infty$. Then the inequality

$$\begin{aligned} \forall \,h\in \mathcal D({\mathbb R}^d),\quad \int _{{\mathbb R}^d}{|\nabla h|^2}\,\mathrm d\xi \ge \mathcal {C}\int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2}\,\mathrm d\nu \end{aligned}$$

(29)

holds for some optimal constant $\mathcal {C}\in (0,\mathcal {C}_\star ]$, where $\widetilde{h}:=\int _{{\mathbb R}^d}{h}\,\mathrm d\xi$. Here $\mathcal {C}_\star$ denotes the optimal constant in (24).

Notice that (29) is similar to (24), except that the average is computed with respect to the measure of the left-hand side. We emphasize that in (29), the right-hand side is not the variance of h with respect to the measure $\mathrm d\nu$, as we subtract the average with respect to the measure $\mathrm d\xi$. In the case $\phi (v)=\left\langle v\right\rangle ^\alpha +\log Z_\alpha$, which corresponds to (25), Inequality (29) is precisely (10). Inequality (10) has been established in [17, inequality (1.12)] by a different method, based on the strategy of [1, 2]. Also see “Appendix B.1” for further details. As we shall see in the proof, our method provides an explicit lower bound $\mathcal {C}$ in terms of $\mathcal {C}_\star$.

Proof

Without loss of generality, we can assume that $\widetilde{h}=\int _{{\mathbb R}^d}{h}\,\mathrm d\xi =0$ up to the replacement of h by $h-\widetilde{h}$. We use the IMS decomposition method (see [24, 28]), which goes as follows. Let $\chi$ be a truncation function on ${\mathbb R}_+$ with the following properties: $0\le \chi \le 1$, $\chi \equiv 1$ on [0, 1], $\chi \equiv 0$ on $[2,+\infty )$ and ${\chi '}^2/\left( 1-\chi ^2\right) \le \kappa$ for some $\kappa >0$. Next, we define $\chi _R(v)=\chi \big (|v|/R\big )$, $h_{1,R}=h\,\chi _R$ and $h_{2,R}=h\,\sqrt{1-\chi _R^2}$, so that $h_{1,R}$ is supported in the ball $B_{2R}$ of radius 2R centered at $v=0$ and $h_{2,R}$ is supported in $B_R^c={\mathbb R}^d\setminus B_R$. Elementary computations show that $h^2=h_{1,R}^2+h_{2,R}^2$ and $|\nabla h|^2=|\nabla h_{1,R}|^2+|\nabla h_{2,R}|^2-h^2\,|\nabla \chi |^2/\left( 1-\chi ^2\right)$, so that $\big ||\nabla h|^2-|\nabla h_{1,R}|^2-|\nabla h_{2,R}|^2\big |\le \kappa \,h^2/R^2$.

Since $h_{2,R}$ is supported in $B_R^c$, we know that

$$\begin{aligned} \int _{{\mathbb R}^d}{|\nabla h_{2,R}|^2}\,\mathrm d\xi \ge \mathsf q(R)\int _{{\mathbb R}^d}{|h_{2,R}|^2}\,\mathrm d\nu \end{aligned}$$

for any $R>0$, where $\mathsf q$ is the quotient involved in the definition (26) of $\sigma _0$. We recall that $\lim _{r\rightarrow +\infty }\mathsf q(r)=\sigma _0>0$. Using the method of the Holley-Stroock lemma (see [9, 15] for a recent presentation), we deduce from inequality (24) that

$$\begin{aligned} \int _{{\mathbb R}^d}{|\nabla h_{1,R}|^2}\,\mathrm d\xi&\ge \mathcal {C}_\star \int _{{\mathbb R}^d}{\left| h_{1,R}-\widehat{h}_{1,R}\right| ^2}\,\mathrm d\nu \\&\ge \mathcal {C}_\star \int _{B_{2R}}\left| h_{1,R}-\widehat{h}_{1,R}\right| ^2\,\psi \,\mathrm d\xi \\&\ge \mathcal {C}_\star \,\inf _{B_{2R}}\psi \,\min _{c\in {\mathbb R}}\int _{B_{2R}}\left| h_{1,R}-c\right| ^2\,\mathrm d\xi \\&\ge \mathsf Q(R)\int _{{\mathbb R}^d}{\left| h_{1,R}\right| ^2}\,\mathrm d\nu -\mathcal {C}_\star \,\frac{\inf _{B_{2R}}\psi }{\xi (B_{2R})}\left( \int _{{\mathbb R}^d}{h_{1,R}}\,\mathrm d\xi \right) ^2 \end{aligned}$$

where $\mathsf Q(R):=\mathcal {C}_\star \,\inf _{B_{2R}}\psi /\sup _{B_{2R}}\psi$. By the assumption $\widetilde{h}=0$, we know that

$$\begin{aligned} \int _{B_R}h\,\mathrm d\xi =-\int _{B_R^c}h\,\mathrm d\xi , \end{aligned}$$

from which we deduce that

$$\begin{aligned} \left( \int _{{\mathbb R}^d}{h_{1,R}}\,\mathrm d\xi \right) ^2&=\left( \int _{B_R}h\,\mathrm d\xi +\int _{B_R^c}\chi _R\,h\,\mathrm d\xi \right) ^2\\&\le \left( \int _{B_R^c}|h|\,\mathrm d\xi \right) ^2\le \int _{{\mathbb R}^d}{h^2}\,\mathrm d\nu \int _{B_R^c}\psi ^{-1}\,\mathrm d\xi \end{aligned}$$

where the last inequality is simply a Cauchy-Schwarz inequality. Let

$$\begin{aligned} \varepsilon (R):=\mathcal {C}_\star \,\frac{\inf _{B_{2R}}\psi }{\xi (B_{2R})}\int _{B_R^c}\psi ^{-1}\,\mathrm d\xi \,. \end{aligned}$$

By the assumption that $\psi ^{-1}\in \mathrm L^1({\mathbb R}^d,\mathrm d\xi )$, we know that

$$\begin{aligned} \lim _{R\rightarrow +\infty }\varepsilon (R)=0\quad \text{ and }\quad \lim _{R\rightarrow +\infty }\frac{\varepsilon (R)}{\mathsf Q(R)}=0\,. \end{aligned}$$

Collecting all our assumptions, we have

$$\begin{aligned} \int _{{\mathbb R}^d}{|\nabla h|^2}\,\mathrm d\xi&\ge \int _{{\mathbb R}^d}{\left( |\nabla h_{1,R}|^2+|\nabla h_{2,R}|^2-\frac{\kappa }{R^2}\,h^2\right) }\,\mathrm d\xi \\&\ge \left( \min \big \{\mathsf Q(R),\mathsf q(R)\big \}-\varepsilon (R)-\frac{\kappa }{R^2}\right) \int _{{\mathbb R}^d}{|h|^2}\,\mathrm d\nu \end{aligned}$$

where $\min \big \{\mathsf Q(R),\mathsf q(R)\big \}-\varepsilon (R)-\kappa /R^2$ is positive for $R>0$, large enough, as follows from the assumptions on $\psi$.

Finally, let us notice that, for any $c\in {\mathbb R}$, we have

$$\begin{aligned} \int _{{\mathbb R}^d}{\left| h-c\,\right| ^2}\,\mathrm d\nu =\int _{{\mathbb R}^d}{h^2}\,\mathrm d\nu -2\,c\int _{{\mathbb R}^d}{h}\,\mathrm d\nu +c^2\ge \int _{{\mathbb R}^d}{\left| h-\widehat{h}\right| ^2}\,\mathrm d\nu \end{aligned}$$

with equality if and only if $c=\widehat{h}=\int _{{\mathbb R}^d}{h}\,\mathrm d\nu$. As a special case corresponding to $c=\widetilde{h}=\int _{{\mathbb R}^d}{h}\,\mathrm d\xi$, we have

$$\begin{aligned} \int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2}\,\mathrm d\nu \ge \int _{{\mathbb R}^d}{\left| h-\widehat{h}\right| ^2}\,\mathrm d\nu \,. \end{aligned}$$

This proves that $\mathcal {C}_\star \ge \mathcal {C}$. $\square$

In the special case of (25), the assumptions of Corollary 10 are not difficult to check. It is also possible to give a slightly shorter proof using the Poincaré inequality on $B_R$ when  (25) holds: see [22, Chapter 6].

Algebraic decay rates for the Fokker–Planck equation

Here we consider simple estimates of the decay rates in the spatially homogeneous case of equation (1), that is, $f(t,x,v)=g(t,v)$ solving the Fokker-Planck equation

$$\begin{aligned} \partial _tg=\mathsf {L}_1 g\,. \end{aligned}$$

(30)

After summarizing the standard approach based on the weak Poincaré inequality (see for instance [17]) in Section B.1, we introduce a new method which relies on weighted $\mathrm L^2$ estimates. As already mentioned, the advantage of weighted Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential ($\alpha >1$) and exponential ($\alpha =1$) local equilibria to sub-exponential local equilibria, with $\alpha \in (0,1)$.

1.1 Weak Poincaré inequality

We assume $\alpha \in (0,1)$ and $\eta \in \big (0,\beta \big )$ with $\beta = 2\left( 1-\alpha \right)$. By a simple Hölder inequality, with $(\tau +1)/\tau =\beta /\eta$, we obtain

$$\begin{aligned}&\int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2}\,\mathrm d\xi =\int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2\,\left\langle v\right\rangle ^{-\eta }\,\left\langle v\right\rangle ^\eta }\,\mathrm d\xi \\&\quad \le \left( \int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2\,\left\langle v\right\rangle ^{-\beta }}\,\mathrm d\xi \right) ^\frac{\tau }{\tau +1} \left( \int _{{\mathbb R}^d}{\left\| h-\widetilde{h}\right\| _{\mathrm L^\infty ({\mathbb R}^d)}^2\,\left\langle v\right\rangle ^{\beta \,\tau }}\,\mathrm d\xi \right) ^\frac{1}{1+\tau }\,. \end{aligned}$$

We assume that $\mathrm d\xi = Z_\alpha ^{-1} e^{-\left\langle v\right\rangle ^\alpha } \mathrm dv$ as in (25) and take $\widetilde{h}:=\int _{{\mathbb R}^d}{h}\,\mathrm d\xi$. Using (10), we end up with

$$\begin{aligned} \forall \,h\in \mathcal D({\mathbb R}^d),\quad \int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2}\,\mathrm d\xi \le \mathcal {C}_{\alpha ,\tau }\left( \int _{{\mathbb R}^d}{|\nabla h|^2}\,\mathrm d\xi \right) ^\frac{\tau }{1+\tau }\,\left\| h-\widetilde{h}\right\| _{\mathrm L^\infty ({\mathbb R}^d)}^\frac{2}{1+\tau }, \end{aligned}$$

(31)

for some explicit positive constant $\mathcal {C}_{\alpha ,\tau }$. We learn from (6) that

$$\begin{aligned} \frac{\mathrm d}{\mathrm dt}\int _{{\mathbb R}^d}{\left| h(t,\cdot )-\widetilde{h}\right| ^2}\,\mathrm d\xi =-\,2\int _{{\mathbb R}^d}{|\nabla _vh|^2}\,\mathrm d\xi \end{aligned}$$

if $g=h\,F$ solves (30), and we also know that $\widetilde{h}$ does not depend on t. By a strategy that goes back at least to [20, Theorem 2.2] and, according to the author of [20], due to D. Stroock, we obtain

$$\begin{aligned} \int _{{\mathbb R}^d}{\left| h(t,\cdot )-\widetilde{h}\right| ^2}\,\mathrm d\xi \le \left( \left( \int _{{\mathbb R}^d}{\left| h(0,\cdot )-\widetilde{h}\right| ^2}\,\mathrm d\xi \right) ^{-\frac{1}{\tau }}+\frac{2\,\tau ^{-1}}{\mathcal {C}_{\alpha ,\tau }^{1+1/\tau }\,\mathcal M}\,t\right) ^{-\tau } \end{aligned}$$

with $\mathcal M=\sup _{s\in (0,t)}\Vert h(s,\cdot )-\widetilde{h}\Vert _{\mathrm L^\infty ({\mathbb R}^d)}^{2/\tau }$, where the Bihari-LaSalle inequality has been employed again. The limitation is of course that we need to restrict the initial conditions in order to have $\mathcal M$ uniformly bounded with respect to t. Since $\eta$ can be chosen arbitrarily close to $\beta$, the exponent $\tau$ can be taken arbitrarily large but to the price of a constant $\mathcal {C}_{\alpha ,\tau }$ which explodes as $\eta \rightarrow \beta _-$.

Note that, with the denomination used in [27, (1.6)], Formula (31) is equivalent to the weak Poincaré inequality

$$\begin{aligned} \forall \,h\in \mathcal D({\mathbb R}^d),\quad \mathcal {C}_{\alpha ,\tau }^{-1}&\int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2}\,\mathrm d\xi \le \tfrac{\tau }{(1+\tau )^{1+1/\tau }}\, r^{-1/\tau }\\&\int _{{\mathbb R}^d}{|\nabla h|^2}\,\mathrm d\xi +r\,\left\| h-\widetilde{h}\right\| _{\mathrm L^\infty ({\mathbb R}^d)}^2, \end{aligned}$$

for all $r>0$. The equivalence of this inequality and (31) is easily recovered by optimizing on $r>0$. It is worth to remark that here we consider $\Vert h-\widetilde{h}\Vert _{\mathrm L^\infty ({\mathbb R}^d)}$ while various other quantities like, e.g., the median can be used in weak Poincaré inequalities.

1.2 Weighted $\mathrm L^2$ estimates

As an alternative approach to the weak Poincaré inequality method of “Appendix B.1”, we can consider for some arbitrary $k>0$ the evolution according to equation (30) of $\int _{{\mathbb R}^d}|h(t,v)|^2\,\left\langle v\right\rangle ^k\,\mathrm d\xi =\int _{{\mathbb R}^d}|h(t,v)|^2\,\left\langle v\right\rangle ^k\,F\,\mathrm dv$ where $\mathrm d\xi$ is as in (25) and $h:=g/F$ solves

$$\begin{aligned} \partial _th=F^{-1}\,\nabla _v\cdot \big (F\,\nabla _vh\big )\,. \end{aligned}$$

Let us compute

$$\begin{aligned} \frac{\mathrm d}{\mathrm dt}\int _{{\mathbb R}^d}|h(t,v)|^2\,\left\langle v\right\rangle ^k\,F\,\mathrm dv+\,2\int _{{\mathbb R}^d}|\nabla _vh|^2\,\left\langle v\right\rangle ^k\,F\,\mathrm dv=-\int _{{\mathbb R}^d}\nabla _v(h^2)\cdot \big (\nabla _v\left\langle v\right\rangle ^k\big )\,F\,\mathrm dv \end{aligned}$$

and observe with $\ell =2-\alpha$ that

$$\begin{aligned} \nabla _v\cdot \left( F\,\nabla _v\left\langle v\right\rangle ^k\right)= & {} \frac{k}{\left\langle v\right\rangle ^4}\left( d+(k+d-2)\,|v|^2-\alpha \,\left\langle v\right\rangle ^\alpha \,|v|^2\right) F\left\langle v\right\rangle ^k \\\le & {} \left( a-b\,\left\langle v\right\rangle ^{-\ell }\right) F\left\langle v\right\rangle ^k, \end{aligned}$$

for some $a\in {\mathbb R}$, $b\in (0,+\infty )$. This estimate corresponds to Lemma 5 for the spatially inhomogeneous equation. From here the same proof as in Proposition 4 shows that there exists a constant $\mathcal K_k>0$ such that

$$\begin{aligned} \forall \,t\ge 0\,\quad \left\| h(t,\cdot )\right\| _{\mathrm L^2\left( \left\langle v\right\rangle ^k\,\mathrm d\xi \right) }\le \mathcal K_k\,\left\| h^\mathrm {in}\right\| _{\mathrm L^2\left( \left\langle v\right\rangle ^k\,\mathrm d\xi \right) }\,. \end{aligned}$$

Hence, if $g= hF$ solves (30) with initial value $h^\mathrm {in}$, we can use (10) to write

$$\begin{aligned} \frac{\mathrm d}{\mathrm dt}\int _{{\mathbb R}^d}{\left| h(t,\cdot )-\widetilde{h}\right| ^2}\,\mathrm d\xi =-\,2\int _{{\mathbb R}^d}{|\nabla _vh|^2}\,\mathrm d\xi \le -\,2\,\mathcal {C}\int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2\,\left\langle v\right\rangle ^{-\beta }}\,\mathrm d\xi \end{aligned}$$

with $\beta = 2\left( 1-\alpha \right)$ and $\widetilde{h}=\int _{{\mathbb R}^d}{h}\,\mathrm d\xi$. With $\theta =k/\big (k+\beta \big )$, Hölder’s inequality

$$\begin{aligned} \int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2}\,\mathrm d\xi \le \left( \int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2\,\left\langle v\right\rangle ^{-\beta }}\,\mathrm d\xi \right) ^\theta \left( \int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2\,\left\langle v\right\rangle ^k}\,\mathrm d\xi \right) ^{1-\theta } \end{aligned}$$

allows us to estimate the right hand side and obtain the following result.

Proposition 11

Let $\alpha \in (0,1)$, let $g^\mathrm {in}\in \mathrm L^1_+(\mathrm d\mu )\cap \mathrm L^2(\left\langle v\right\rangle ^k\mathrm d\mu )$ for some $k>0$, and consider the solution g to (30) with initial datum $g^\mathrm {in}$. With $\mathcal {C}$ as in (10), if $\overline{g}=\left( \int _{{\mathbb R}^d}g\,\mathrm dv\right) F$ where $F$ is given by (2), then

$$\begin{aligned} \int _{{\mathbb R}^d}{\left| g(t,\cdot )-\overline{g}\right| ^2}\,\mathrm d\mu \le \left( \left( \int _{{\mathbb R}^d}{\left| g^\mathrm {in}-\overline{g}\right| ^2}\,\mathrm d\mu \right) ^{-\beta /k}+\frac{2\,\beta \,\mathcal {C}}{k\,\mathcal K^{\beta /k}}\,t\right) ^{-k/\beta } \end{aligned}$$

with $\beta =2\,(1-\alpha )$ and $\mathcal K:=\mathcal K_k^2\,\left\| g^\mathrm {in}\right\| _{\mathrm L^2\left( \left\langle v\right\rangle ^k\,\mathrm d\mu \right) }^2+\Theta _k\left( \int _{{\mathbb R}^d}g^\mathrm {in}\,\mathrm dv\right) ^2$.

We recall that $g=h\,F$, $\,\overline{g}=\widetilde{h}\,F$ and $F\,\mathrm d\mu = \mathrm dv = F^{-1}\,\mathrm d\xi$. We note that arbitrarily large decay rates can be obtained under the condition that $k>0$ is large enough. We recover that when $k<d\,\beta /2$, the rate of relaxation to the equilibrium is slower than $(1+t)^{-d/2}$ and responsible for the limitation that appears in Theorem 1. However, the rate of the heat flow is recovered in Theorem 1 for a weight of order k with an arbitrarily small but fixed $k>0$, if $\alpha$ is taken close enough to 1.

Proof

Using

$$\begin{aligned} \int _{{\mathbb R}^d}{\left| h-\widetilde{h}\right| ^2\,\left\langle v\right\rangle ^k}\,\mathrm d\xi \le \int _{{\mathbb R}^d}{|h|^2\,\left\langle v\right\rangle ^k}\,\mathrm d\xi +\Theta _k\,\widetilde{h}^2=\mathcal K, \end{aligned}$$

we obtain that $y(t):=\int _{{\mathbb R}^d}{\left| g(t,\cdot )-\overline{g}\right| ^2}\,\mathrm d\mu$ obeys to $y'\le -\,2\,\mathcal {C}\,\mathcal K^{1-1/\theta }\,y^{1/\theta }$ and conclude by the Bihari-LaSalle inequality. $\square$