Convergence to Equilibrium in Energy-Reaction–Diffusion Systems Using Vector-Valued Functional Inequalities

Abstract

We discuss how the recently developed energy dissipation methods for reaction diffusion systems can be generalized to the non-isothermal case. For this, we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions, we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

Appendices

Generalized EEP Estimates

This appendix gives a proof of the EEP estimate (3.6) for general $\gamma$ and $\alpha$. For a more detailed exposition, we refer to Mittnenzweig (2014). We will use the Poincaré-Sobolev inequality

$$\begin{aligned} \left\| \nabla u\right\| _{p}\ge \eta _{\text {PS}}\left( \Omega ,p,q\right) \left\| u {-}\overline{u}\right\| _{q} \end{aligned}$$

(A.1)

corresponding to the Sobolev embedding $W^{1,p}\left( \Omega \right) \hookrightarrow L^{q}\left( \Omega \right)$, where $\eta _{\text {PS}}\left( \Omega ,p,q\right)$ is the optimal Poincaré-Sobolev constant. This inequality holds true for $p\ge d$ or for $p<d$ and $q\le {dp}/(d{-}p).$

Theorem A.1

Let $\gamma >\max \big \{0, 1{-}2/{d}\big \}$ and $\alpha \le \gamma {+} {2}/{d}$ for $d\ge 3$ or $\alpha <1{+}\gamma$ for $d\le 2$. Then $\rho (\Omega ,\gamma ,\alpha )$ is strictly positive and can be bounded by

$$\begin{aligned} \rho (\Omega ,\gamma ,\alpha )\ge \frac{4\alpha \big (\gamma {-} {2}/{q}\big )}{\gamma ^{2}\left( q{-}1\right) } \; \eta _{\text {PS}}\left( \Omega ,2,q\right) ^2 \end{aligned}$$

(A.2)

with $q\ge \max \big \{ {2}/{\gamma },{2}/(1{+} \gamma {-}\alpha )\big \}$.

Proof

We substitute $v=u^{\gamma /2}$. Then

$$\begin{aligned} \overline{u}^{\gamma }\int _{\Omega }F_{\alpha _{1}}\left( \frac{u}{\overline{u}}\right) \;\!\mathrm {d}x&\le \frac{\alpha _{2}}{\alpha _{1}}\overline{u}^{\gamma } \int _{\Omega }F_{\alpha _{2}}\left( \frac{u}{\overline{u}}\right) \;\!\mathrm {d}x =C\left( \frac{\left\| v\right\| _{x}^{x}}{\left\| v \right\| _{y}^{x-2}}-\left\| v\right\| _{y}^{2}\right) \\&\le C(\left\| v\right\| _{z}^{2}{-}\left\| v\right\| _{y}^{2})\le C\left( z{-}1\right) \left\| v{-}\overline{v}\right\| _{z}^{2} \le \frac{C\left( z{-}1\right) }{\eta _{\text {PS}} \left( \Omega ,2,z\right) ^2}\left\| \nabla v\right\| _{2}^{2} \end{aligned}$$

with $\alpha _{2}>1$, $C={1}/({\alpha _{1}\left( \alpha _{2}{-}1\right) }),$ $x={2\alpha _{2}}/{\gamma }$, $y={2}/{\gamma }$, and $z={2}/(1{+}\gamma {-} \alpha _{2})$. In the first line we used monotonicity of the map $\alpha \mapsto \alpha F_{\alpha }\left( z\right)$, see (3.2). In the last line, we applied the Hölder inequality $\left\| v\right\| _{x}^{x}\le \left\| v\right\| _{y}^{x-2}\left\| v\right\| _{z}^{2}$, the Poincaré–Sobolev inequality and the following estimate which holds true for $x\ge 2$ (see Mittnenzweig (2014) for a proof):

$$\begin{aligned} \left\| v\right\| _{x}^{2}-\left\| v\right\| _{y}^{2}\le \left\| v\right\| _{x}^{2}-\left\| v\right\| _{1}^{2}\le \left( x{-}1\right) \left\| v {-}\overline{v}\right\| _{x}^{2} \end{aligned}$$

This proves the theorem. $\square$

Remark A.2

The above theorem still remains true for the cases $\gamma =1-{2}/{d}$ as well as $\alpha =\gamma +1$ for $d\le 2$, see Mittnenzweig (2014). In the case $\alpha =2$ and $\gamma =1$, which will be used below, we have the simple bound

$$\begin{aligned} \rho (\Omega ,1,2) \ge 2 \, \eta _{\text {PS}} (\Omega ,1,2)^2, \end{aligned}$$

(A.3)

which follows from the following chain of estimates:

$$\begin{aligned} \overline{u} \int _{\Omega } \frac{|\nabla u |^2}{u} \;\!\mathrm {d}x {\ge } \left\| \nabla u \right\| _1^2 {\ge } \eta _\text {PS} (\Omega ,1,2)^2 \left\| u{\!-\!} \overline{u} \right\| ^2 {\!=\!} 2 \, \eta _\text {PS} (\Omega ,1,2)^2 \cdot \overline{u}^{2}\int _{\Omega }F_{2} (u/\overline{u})\;\!\mathrm {d}x. \end{aligned}$$

The EEP Estimate for $\alpha =\gamma =0$ and $d\le 2$

This appendix provides a proof that the EEP inequality also holds in the case $\alpha =\gamma =0$. Here we only give a densified version of the full proof and refer to Mittnenzweig (2014) for a detailed exposition, which is based on the approach developed in Carlen and Loss (1992) and Carlen et al. (2010). We note that $\rho (\Omega , 1,2)$ is positive only for $\Omega \subset {\mathbb R}^d$ with $d \le 2.$

Theorem B.1

(Case $\alpha =\gamma =0$ in dimension $d\le 2$) We have the estimate $\rho (\Omega ,0,0)\ge \varrho :=\rho (\Omega ,1,2)>0$, i.e.,

$$\begin{aligned} \forall \ u>0:\quad \int _\Omega \frac{|\nabla u(x)|^2}{u(x)^2} \;\!\mathrm {d}x \ge \rho (\Omega ,1,2) \int _\Omega F_0(u(x)/\mathsf U)\;\!\mathrm {d}x. \end{aligned}$$

(B.1)

Proof

We set $u=\mathrm {e}^v$ and observe that (B.1) is equivalent to

$$\begin{aligned} H(v):=\int _\Omega |\nabla v|^2\;\!\mathrm {d}x \ge \varrho G(v) \text { with }G(v):=\Big ( \log \Big (\int _\Omega \mathrm {e}^v\;\!\mathrm {d}x\Big ) - \int _\Omega v \;\!\mathrm {d}x \Big ). \end{aligned}$$

(B.2)

We consider H and G as nonnegative, proper, and convex functionals on ${\mathrm L}^2(\Omega )$. Because of $\Omega \subset {\mathbb R}^d$ with $d\le 2$, the Moser–Trudinger inequality G is bounded whenever H is bounded.

We establish (B.2) by Legendre transform for which we note

$$\begin{aligned} H^*(\psi )=\frac{1}{4}\int _\Omega (\psi {-}\overline{\psi })({-}\Delta )^{-1}(\psi {-}\overline{\psi })\;\!\mathrm {d}x \text { and } G^*(\psi )=\left\{ \begin{array}{cl}\int _\Omega \Gamma (\psi (x))\;\!\mathrm {d}x &{}\text { if }\overline{\psi }=0,\\ \infty &{}\text {else}, \end{array}\right. \end{aligned}$$

where $\Gamma (\psi )=\lambda _{\mathfrak B}(\psi {+}1)$ for $\psi \ge -1$ and $\infty$ otherwise.

We now define $J(\psi )=G^*(\psi )-\varrho H^*(\psi )$ and show $J(\psi _0)\ge 0$ for all $\psi _0$. For this, we note $J(0)=0$ and then consider J along the solutions $\psi _t:=\mathrm {e}^{t\Delta }\psi _0$ of the diffusion equation $\dot{\psi } = \Delta \psi$. Setting $\psi _t =c_t{-}1$ with $\overline{c}_t=1$ and $c_t\ge 0$. By differentiating along solutions and using $\dot{c}_t =\Delta c_t$ (with Neumann boundary conditions), we obtain

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}t} J(c_t{-}1) = -\int _\Omega \frac{|\nabla c_t|^2}{c_t}\;\!\mathrm {d}x + \frac{\varrho }{2}\int _\Omega (c_t{-}1)^2\;\!\mathrm {d}x \ge 0, \end{aligned}$$

since $\varrho =\rho (\Omega ,1,2)$ which is the optimal constant for the last estimate. Hence, $J(\psi _0)\ge J(\psi _t) \ge J(0)=0$ since $\psi _t\rightarrow 0$.

Now, we reverse the Legendre transform using $H^*(\varrho \psi )=\varrho ^2 H^*(\psi )$, hence

$$\begin{aligned} H(v)&=\ \sup \big ( \langle v,\psi \rangle {-}H^*(\psi )\big ) \ \overset{\psi =\varrho \widetilde{\psi }}{=} \ \sup \big ( \langle v, \varrho \widetilde{\psi }\rangle {-}\varrho ^2 H^*(\widetilde{\psi })\big )\\&\overset{J\ge 0}{=} \varrho \sup \big ( \langle v,\widetilde{\psi }\rangle {-}G^*(\widetilde{\psi })\big ) \ = \ \varrho G(v). \end{aligned}$$

This proves the assertion. $\square$