## Entropy method for chemical reaction-diffusion systems I: Entropy method (continued)

In the last post, we’ve already seen the motivation of the question of convergence to equilibrium. This post continues to give a method which does not only give qualitative result (does the trajectory converge to equilibrium?) but (usually) also quantitave result (how fast is the convergence?).

Gronwall’s inequality

Let us start with the classic Gronwall’s inequality: Assume that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function. If

$\boxed{\partial_tf(t) \leq \lambda f(t)}$ for all $t\geq 0$

then

$\boxed{f(t) \leq e^{\lambda t}f(0)}$ for $t\geq 0$.

Especially, when $\lambda < 0$ then $f(t)$ decays exponentially.

A heat equation

We continue to see how to show the convergence to equilibrium for a heat equation with homogeneous Neumann boundary condition: let $\Omega \subset \mathbb{R}^N$ be a bounded domain with $C^1$ boundary. Consider the heat equation for $u(x,t)$

$u_t - \Delta u = 0$  in $\Omega$

$\partial_{\nu} u = 0$ on $\partial\Omega$

$u(x,0) = u_0(x)$ in $\Omega$.

The equilibrium $u_{\infty}$ satisfies $-\Delta u_{\infty} = 0$ in $\Omega$ and $\partial_{\nu}u_{\infty} = 0$ on $\partial \Omega$. Then we see that any constant function $u_{\infty} \equiv constant$ is an equilibrium.

However, note that from $u_t - \Delta u = 0$, we get, after integrating over $\Omega$

$\partial_t \int_{\Omega}udx - \int_{\Omega}\Delta u dx = 0$.

Using the homogeneous Neumann boundary condition we have $\int_{\Omega} \Delta u dx = 0$. Then it follows that

$\partial_t \int_{\Omega} udx = 0$

and consequently

$\int_{\Omega} u(x,t)dx = \int_{\Omega}u_0(x)dx$    for all $t>0$.

This is called the mass conservation (or conservation of mass) of the heat equation. This gives us a hint that the equilibrium should also satisfies the mass conservation, i.e.

$\int_{\Omega} u_{\infty}dx = \int_{\Omega} u_0(x)dx$ or equivalently $\boxed{u_{\infty} = \frac{1}{|\Omega|}\int_{\Omega}u_0(x)dx}$.

We now aim to prove the convergence of $u(x,t)$ to $u_{\infty}$ as $t\rightarrow +\infty$.

Denote by $\|\cdot\|$ the usual norm of $L^2(\Omega)$. We define an entropy functional

$E(u) = \|u(t) - u_{\infty}\|^2$.

Compute the time derivative of $E(t)$ we have

$\frac{d}{dt}E(u) = 2\int_{\Omega}(u - u_{\infty})u_tdx = 2\int_{\Omega}(u-u_{\infty})\Delta u dx = -2\|\nabla u\|^2$.

We call the quantity $D(u) := -\frac{d}{dt}E(u) = 2\|\nabla u\|^2$ the entropy dissipation (or entropy production). By the Poincaré inequality

$\|\nabla u\|^2 \geq \lambda \left\|u - \frac{1}{|\Omega|}\int_{\Omega}udx\right\|^2$

we have

$D(u) = 2\|\nabla u\|^2 \geq 2\lambda \left\|u - \frac{1}{|\Omega|}\int_{\Omega}udx\right\|^2 = 2\lambda\|u - u_{\infty}\|^2 = 2\lambda E(u)$.

This is called an entropy-entropy dissipation estimateTherefore, we have

$\frac{d}{dt}E(u) = -D(u) \leq -2\lambda E(u)$,

thus, by the Gronwall lemma

$E(u)(t) \leq e^{-2\lambda t}E(u)(0)$

or equivalently

$\|u(t) - u_{\infty}\| \leq e^{-\lambda t}\|u_0 - u_{\infty}\|$.

Theorem. The trajectory $u(x,t)$ of the heat equation with homogeneous Neumann boundary condition converges exponentially to $u_{\infty}$ with the rate $\lambda$ where $\lambda$ is the constant in the Pointcaré inequality.

Entropy method

We now state the basic idea of the entropy method for an evolution equation of the form

$\partial_t f = F(f)$

which has a unique equilibrium $f_{\infty}$ and a set of mass conservations.

We aim to find

(i) an entropy functional $E(f)$ which has the property $E(f) - E(f_{\infty}) \geq C\|f - f_{\infty}\|$ and has

(ii) nonnegative entropy dissipation $D(f) = -\frac{d}{dt}E(f) \geq 0$; and

(iii) an entropy-entropy dissipation estimate of the form

$D(f) \geq \lambda(E(f) - E(f_{\infty}))$.

If we have (i)-(iii) then it follows first by Gronwall’s lemma that $E(f)(t) \rightarrow E(f_{\infty})$ exponentially and then by (i) that $f(t) \rightarrow f_{\infty}$ exponentially.

(Some) Good things about entropy method

(i) The entropy method is based on an functional inequality (entropy-entropy dissipation estimate) which is usually not directly linked to the evolution system. That makes the method is quite robust to generalisation. Once the functional inequality is established, it can be used in any other system which has similar kind of entropy functional and entropy dissipation.

(ii) The entropy method usually gives explicit rate of convergence. If we can prove the functional inequality with explicit constant, then the convergence follows with explicit constant. This is the quantitative result that has been mentioned before.