On Uniform Decay of the Entropy for Reaction–Diffusion Systems

Abstract

This work provides entropy decay estimates for classes of nonlinear reaction–diffusion systems modeling reversible chemical reactions under the detailed-balance condition. We obtain explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the Log-Sobolev estimate and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: (i) vanishing diffusion constants in some chemical components and (ii) usage of different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

Appendices

Appendices

1.1 Proof of Lemma 4.3

Proof

To derive the lower bound (4.6) we use $F_1(z)\ge \frac{1}{2} F_{1/2}(z)= (\sqrt{z}{-}1)^2$ and ${\mathbb {G}}(u,v)\ge 4 (v{-}\sqrt{u})^2$ to obtain $F_1(u)+\kappa {\mathbb {G}}(u,v) \ge (\sqrt{u}{-}1)^2+4 \kappa (v{-}\sqrt{u})^2 \ge \tfrac{4 \kappa }{1+4 \kappa }(v{-}1)^2$, where we minimized explicitly with respect to $\sqrt{u}$. The estimate follows via $(z {-} 1)^2=2 F_2(z)\ge F_1(z)$.

For the analysis of the convexity the basic observation is that $\phi _\kappa$ has the scaling property

$$\begin{aligned} \phi _\kappa ( s^2 u, s v) = s^2 \phi _\kappa (u,v) + 2 s^2 u \log s + 1 - s^2, \end{aligned}$$

which follows from ${\mathbb {G}}(s^2 u, s v)=s^2 {\mathbb {G}}(u,v)$ and (3.3). Since this scaling is affine in $(u,v,\phi )$ (for each $s$), this property is inherited by the convexification $\phi ^{**}_\kappa$. Introducing $h_\kappa (z)=\phi _\kappa (z,1)$ and $g_\kappa (z)=\phi ^{**}_\kappa (z,1)$, we have the representations

$$\begin{aligned} \phi _\kappa (u,v)&= v^2 h_\kappa (u/v^2) + 2u\log v + 1 - v^2, \end{aligned}$$

(6.1)

$$\begin{aligned} \phi _\kappa ^{**}(u,v)&= v^2 g_\kappa (u/v^2) + 2u\log v + 1 - v^2, \end{aligned}$$

(6.2)

A direct calculation of ${\mathrm {D}}^2\phi _\kappa ^{**}$ shows that $\phi _\kappa ^{**}$ defined as in (6.2) is convex if and only if

$$\begin{aligned} g''_\kappa (z)\ge 0 \text { and } g''_\kappa (z)\big ( g_\kappa (z) - zg'_\kappa (z) +3 z -1) \ge 2. \end{aligned}$$

(6.3)

Inserting $h_\kappa : z\mapsto F_1(z) + \kappa (z{-}1)\log z$ into this criterion one obtains convexity of $\phi _\kappa$ for $\kappa \in [0,\kappa _*]$ with $\kappa _*\approx 0.8564998142$, where at $z\approx 12.683$ the second criterion in (6.3) holds with equality. In fact, for $h_\kappa$ the second criterion in (6.3) can be rewritten, after dividing by $\kappa >0$, in the form $z^2+3z - z \log z \ge \kappa \big ( z^2 -1 +(1{+}z) \log z\big )$ for all $z>0$. Using $\log z \le \log 3 +z/3-1$ it is easy to see that the estimate holds for $\kappa =1/2$, and we conclude $\kappa _*\ge 1/2$. Moreover, since the right-hand side is positive for all $z>0$ while the left-hand side is negative for $z<1$, we have the explicit characterization

$$\begin{aligned} \kappa _* {:}=\inf \{\, \tfrac{z^2+3z - z \log z}{ z^2 -1 +(1{+}z) \log z} \, | \, z>1 \,\} . \end{aligned}$$

For $\kappa \ge 1$ we obtain an upper bound on $g_\kappa$ by estimating $g_\kappa (z)=\phi ^{**}_\kappa (z,1)$ with $(1{-}\theta ) \phi _\kappa (0,0) + \theta \phi _\kappa ( z/\theta ,1/\theta )$. Using $\phi _\kappa (0,0)=1$, (6.1), and defining $r=1/\theta >1$, yields

$$\begin{aligned}&g_\kappa (z) \le r h_\kappa (z/r)+ 2 z \log r +1-r = h_\kappa (z) + \beta _\kappa (r,z).\\&\text { with } \beta _\kappa (r,z)=(rk{-}kz{+}z)\log r - k(r {-} 1) \log z. \end{aligned}$$

Analyzing $\beta _\kappa (\cdot ,z)$ on the interval ${[1,\infty [}$, we find the following: Define $Z_\kappa$ as in the statement of the lemma (where we use $\kappa \ge 1$); then $Z_1=\mathrm {e}$ and $\kappa \mapsto Z_\kappa$ decreases monotonously with $Z_\infty =1$. For $z\le Z_\kappa$ we have $\beta _\kappa (r,z)\ge \beta _\kappa (1,z)=0$. For $z> Z_\kappa$ the unique minimizer $r$ of $\beta _\kappa (\cdot ,z)$ is given as $r=z/Z_\kappa$. Thus, for $z> Z_\kappa$ we have $g_\kappa ( z) \le h_\kappa (z) + \beta _\kappa ( z/Z_\kappa , z) = 2 z \log z -B_\kappa +1$, with $B_\kappa$ as given in the statement of the lemma. Note that $B_1=1+1/\mathrm {e}$ and that $\kappa \mapsto B_\kappa$ decays monotonously with $B_\infty =1$.

It remains to be shown that $\phi _\kappa ^{**}$ in (6.2) with $g_\kappa$ defined in (4.8) is convex. For this we check (6.3). For $z\ge Z_\kappa$ this is immediate, since the second estimate holds with equality. For $z\le Z_\kappa$ we consider $h_\kappa$ and obtain

$$\begin{aligned} a_\kappa (z){:}=h''_\kappa \big ( h_\kappa -z h''_\kappa +3z-1)-2=\frac{\kappa }{z^2} \Big ( \kappa - \kappa z^2 + 3z + z^2 - \big (\kappa + z +\kappa z \big )\log z\Big ). \end{aligned}$$

For $z\le 1$ we immediately have $a_\kappa (z)>0$. For $z>1$ we can rearrange to

$$\begin{aligned}&a_\kappa (z)= \frac{\kappa }{z^2}\,(1{+}z) (z{+}\log z {-}1)\big ( f_1(z)f_2(z) -\kappa \big ), \ \ f_1(z)=\tfrac{z}{z{+}\log z {-}1}, \ \ f_2(z)=\tfrac{3{+}z{-}\log z}{ 1{+}z}. \end{aligned}$$

We have $f_2(z)\ge \frac{2{+}\mathrm {e}}{1{+}\mathrm {e}} > 1.268$ for $z\in [1,\mathrm {e}]$. Moreover $f'_1(z)<0$ for $z>1$ which gives, for all $z\in [1,Z_\kappa ]$ the estimate $f_1(z) \ge f_1(Z_k)= \kappa$. Using $Z_\kappa \le \mathrm {e}$ we obtain $f_1(z)f_2(z)-\kappa \ge 0.268 \kappa >0$ and conclude $a_\kappa (z)>0$. Thus, the convexity of $\phi _\kappa ^{**}$ given in (4.8) is established, and Lemma 4.3 is proved. $\square$

1.2 Proof of Lemma 4.6

Proof

We have to show that for each ${\varvec{a}}=(a,b)$ there exists $\Theta ({\varvec{a}})$ such that $\varPhi ^{**}_\theta ({\varvec{a}})=\varPhi _\theta ({\varvec{a}})$ for $\theta \in [0,\Theta ({\varvec{a}})]$. The argument relies on the fact that for $\theta < 1-k/(r\kappa _*)$ the function ${\varvec{c}}\mapsto (1{-}\theta ) r F_1(u)+ k {\mathbb {G}}({\varvec{c}})$ is strictly convex and coercive. Hence subtracting $\theta F_2(v)$ with sufficiently small $\theta$ produces a function that still coincides with its lower convex hull in a large region. To be more precise, we have to show that $\varPhi _\theta$ lies above its Taylor polynomial ${\mathrm {T}}^1_{\varvec{a}}\varPhi _\theta$ of first order expanded in ${\varvec{a}}$: $\big ({\mathrm {T}}^1_{\varvec{a}}\varPhi _\theta \big )({\varvec{c}}){:}= \varPhi _\theta ({\varvec{a}})+ {\mathrm {D}}\varPhi _\theta ({\varvec{a}}) \cdot ({\varvec{c}}{-}{\varvec{a}})$. For this we use special relations for the entropy functions $F_0$ and $F_1$, namely

$$\begin{aligned} F_1(z) = \big ({\mathrm {T}}^1_w F_1\big )(z) + w F_1(z/w) \quad \text {and} \quad F_0(z)=\big ({\mathrm {T}}^1_w \big )F_0(z) + F_0(z/w). \end{aligned}$$

For ${\mathbb {G}}(u,v)=\varGamma (u,v^2)=v^2 \varGamma (u/v^2,1)$ we use $\varGamma (z,1)=F_0(z)+F_1(z)$ and obtain

$$\begin{aligned} {\mathbb {G}}(u,v)\!=\! v^2 \Big ( F_1\big (\tfrac{a}{b^2}\big ) {+}F_0\big (\tfrac{a}{b^2}\big )\Big ) \!+\! \Big ( F'_1\big (\tfrac{a}{b^2}\big ){+}F'_0\big (\tfrac{a}{b^2}\big ) \Big ) \big ( u - \tfrac{av^2}{b^2}\big ) \!+\! v^2\Big ( \tfrac{a}{b^2}F_1\big (\tfrac{ub^2}{v^2a}\big ) {+}F_0\big (\tfrac{ub^2}{v^2a}\big ) \Big ). \end{aligned}$$

Note that $F_0$, $F_1$, and ${\mathbb {G}}(\cdot ,v)$ are convex functions, hence the last term in the expansion, which represents the remainder with respect to the first-order Taylor polynomial, is nonnegative and vanishes at ${\varvec{c}}={\varvec{a}}$. Thus, we decompose of the remainder ${\mathcal {R}}$ for $\varPhi _\theta$:

$$\begin{aligned}&{\mathcal {R}}_{\varvec{a}}({\varvec{c}}){:}=\tfrac{1}{k}\Big (\varPhi _\theta ({\varvec{c}})- \big ({\mathrm {T}}^1_{\varvec{a}}\varPhi _\theta \big )({\varvec{c}})\Big )= M({\varvec{a}},v)+ N_{(1-\theta )\rho }({\varvec{a}},u,v) - \theta \rho b F_1(\tfrac{v}{b}) \quad \text {with}\\&M({\varvec{a}},v){:}= \big (\tfrac{v^2}{b^2}-1\big ) \big ( {\mathbb {G}}({\varvec{a}}){-} a {\mathrm {D}}_a {\mathbb {G}}({\varvec{a}})\big ) - {\mathrm {D}}_b {\mathbb {G}}({\varvec{a}})(v{-} b) =\big ( {\mathbb {G}}({\varvec{a}}) {-} a {\mathrm {D}}_a {\mathbb {G}}({\varvec{a}})\big )\big (\tfrac{v}{b}{-}1 \big )^2 \\&N_{\widetilde{\rho }}({\varvec{a}},u,v){:}= \widetilde{\rho }a F_1(\tfrac{u}{a}) + v^2\Big ( \tfrac{a}{b^2}F_1\big (\tfrac{ub^2}{v^2a}\big ) {+}F_0\big (\tfrac{ub^2}{v^2a}\big ) \Big ) \ge 0. \end{aligned}$$

To show positivity for all ${\varvec{c}}$ we first minimize $N_\theta ({\varvec{a}},u,v)$ with respect to the convex variable $u$. The assumption $\rho =1/\kappa > 1/\kappa _*$ and the convexity of $\phi _\kappa$ (cf. Lemma 4.3) yield $M({\varvec{a}},v)+N_\theta ({\varvec{a}},{\varvec{c}}) \ge 0$. Setting $\widehat{n}(\widetilde{\rho },{\varvec{a}},v)= \min \{\, N_{\widetilde{\rho }} ({\varvec{a}},u,v) \, | \, u>0 \,\} \ge 0$ we obtain $M({\varvec{a}},v)+N_{\widetilde{\rho }}({\varvec{a}},{\varvec{c}})\ge M({\varvec{a}},v)+\widehat{n}(\widetilde{\rho },{\varvec{a}},v)$. Thus, we have ${\mathcal {R}}_{\varvec{a}}({\varvec{c}})\ge 0$ if $\rho \theta \in [0,\Theta ((1{-}\theta )\rho ,{\varvec{a}})]$, where

$$\begin{aligned} \Theta (\widetilde{\rho }, {\varvec{a}}){:}= \inf \Big \{\, \frac{M({\varvec{a}},v)+ \widehat{n}(\widetilde{\rho },{\varvec{a}},v)}{ b F_1(v/b)} \,\Big | \, v\ge 0, \ v\ne b \, \Big \}. \end{aligned}$$

Since $\widehat{n}\ge 0$ and $M({\varvec{a}},v)$ grows quadratically with $v$, the infimum is achieved at a finite value of $v$. Since nominator and denominator are smooth functions and strictly positive for $v\ne b$, it suffices to control the behavior for $v \rightarrow b$. Writing $w=a/b^2$ and $v=b{+}\delta$ we have

$$\begin{aligned} M({\varvec{a}},v)\!=\! \mu (w) \delta ^2 \text { with }\mu (w)\!=\!1 {-} \log w {-} w \quad \text {and} \quad \widehat{n}({\varvec{a}},v)\!=\!\frac{2(1{+}w) (\widetilde{\rho }w)}{1+w+\widetilde{\rho }w} \delta ^2 + O(|\delta |^3). \end{aligned}$$

We see that $\mu (w)+\frac{2(1{+}w) (\widetilde{\rho }w)}{1+w+\widetilde{\rho }w}>0$ for all $w>0$ if and only if $\widetilde{\rho }>1/\kappa _*$ with $\kappa _*$ defined in Lemma 4.3. Thus, we have proved $\Theta (\widetilde{\rho },{\varvec{a}})>0$, but without an explicit lower bound.

From the definition of $\widehat{n}$ via $N_{\widetilde{\rho }}$ it is clear that $\partial _{\widehat{\rho }} \widehat{n}\ge 0$, which implies the monotonicity of $\Theta (\cdot ,{\varvec{a}})$. Scaling arguments give $M({\varvec{a}},v)+ \widehat{n}({\varvec{a}},v)= b^2\big ( M(a/b^2,1,v/b)+ \widehat{n}(a/b^2,1,v/b)\big )$. By scaling the denominator as well leads to $\Theta (\widetilde{\rho },a,b)= b \Xi (\widetilde{\rho },a/b^2) \text { with} \Xi (\widetilde{\rho },w)=\Theta (\widetilde{\rho },w,1).$

From $N_{\widetilde{\rho }} \ge 0$ we easily see

$$\begin{aligned} \Xi (\widetilde{\rho },w) \ge \inf \{\, M(w,1,v)/F_1(v) \, | \, v>0 \,\} = \mu (w) \inf \{\, (v{-}1)^2/F_1(v) \, | \, v>0 \,\} = \mu (w), \end{aligned}$$

which gives a positive lower bound for $w<1$. To see the behavior for $w\ge 1$ we express $\widehat{n}$ in terms of the Legendre transform $\gamma _*$ of $\gamma (z)=(z{-}1)\log z$, i.e. $\gamma _*(\zeta )=\sup \{\, z \zeta - \gamma (z) \, | \, z>0 \,\}$. Obviously $\gamma _* \in \mathrm C^\infty ({\mathbb {R}};{\mathbb {R}})$ with $\gamma '_*(\zeta ) >0$ everywhere. The behavior is

$$\begin{aligned} \gamma _*(\zeta )\approx -\log |\zeta | \text { for }\zeta \ll 1,\quad \gamma _*(0)=0,\quad \gamma _*(\zeta )\approx \mathrm {e}^{\zeta } \text { for }\zeta \gg 1. \end{aligned}$$

Using the scaling laws for $F_1$ and $F_0$, see (3.4), we have

$$\begin{aligned} \alpha F_1(s)+F_0(s)=F_1(\alpha s) + F_0(\alpha s) - s(\alpha \log a +\alpha -1) + \alpha + \log \alpha -1 \end{aligned}$$

and conclude $\min \{\, \alpha F_1(s){+} F_0(s) {-} \beta s \, | \, s>0 \,\} = \alpha + \log \alpha -1 - \gamma _*\big ( \beta /\alpha {+} \log \alpha {+} 1 {-} 1/\alpha \big )$. We use this expression for $N_{\widetilde{\rho }}$ with $b=1$, $u=a sv^2$ and $\alpha =(1{+}\widetilde{\rho })a$ to obtain

$$\begin{aligned} \widehat{n}(\widetilde{\rho },a,1,v)= v^2\Big (\alpha {+}\log \alpha -1 - \gamma _*\big (1{+}\log \alpha -1/\alpha - \widehat{\rho }\, \log v^2 \big ) \Big )-\alpha \widehat{\rho }(v^2{-}1) \end{aligned}$$

with $\widehat{\rho }=\tfrac{\widetilde{\rho }}{1+\widetilde{\rho }} > \frac{1}{1+\kappa _*}$. Now the infimum $\Xi (\widetilde{\rho },a)$ can be found numerically by minimizing $\big (\mu (a)(v{-}1)^2+\widehat{n}(\widetilde{\rho },a,1,v)\big )/F_1(v)$ with respect to $v>0$.

Moreover, the function $\xi (\rho ){:}=\inf \{\, \Xi (\rho ,a) \, | \, a>0 \,\}$ can be obtained directly by minimizing $h(a,\rho ,v,w)=(1{-}a{-} \log a) (v{-}1)^2 + a\rho F_1(w v) + v^2(a F_1(w/v)+F_0(w/v))$ with respect to $a>0$ first. Using $(v{-}1)^2 = F_1(w v)- 2 v F_1(w) + v^2 F_1(w/v)$ we obtain

$$\begin{aligned} H(\rho ,v,w){:}=\inf _{a>0} h(a,\rho ,v,w)= (v{-}1)^2\Big (2 + \log \big ( \tfrac{2v F_1(w)+ (\rho {-}1)F_1(wv)}{ (v{-}1)^2}\big ) \Big ) + v^2 F_0(\tfrac{w}{v}) \end{aligned}$$

and conclude

$$\begin{aligned} \xi (\rho ) = \inf \Big \{\, \frac{H(\rho ,v,w)}{F_1(v)} \,\Big | \, 1\ne v>0,\ w>0 \, \Big \}. \end{aligned}$$

Clearly $\xi (1/\kappa _*)=0$, and numerically we find $\xi (1/\kappa _*)=0$, $\xi (1.36976)\approx 1.0$, $\xi (1.5)=1.3038$, $\xi (2)=1.99374$, and $\xi (3)=2.669$. This concludes the proof of Lemma 4.6. $\square$