Entropy method for chemical reaction-diffusion systems I: Entropy method

The concept “entropy method” could mean different things. In this serie of posts, I will address an entropy method which is very  useful in proving convergence to equilibrium evolution equations. The main application will be in chemical reaction network theory.

Convergence to equilibrium of evolution equation

Consider a typical evolution

(1)                              $\boxed{\partial_tf(t) = \mathcal{F}(f)(t)}$         for all            $t>0$

with initial data $f(0) = f_0$. A state $f_{\infty}$ is called an equilibrium if it is a time-independent solution of (1), that is

$\boxed{\mathcal F(f_{\infty}) =0}$.

Many physical or chemical phenomena tend to a stable state in large time. A typical example is a pendulum. After moving it from its equilibrium position, the pendulum starts to move but eventually comes back to its equilibrium because of air friction.

It is noted however that there are also different scenarios like pattern formation or chaos.

Hence, it is interesting to ask whether the trajectory $f(t)$ of (1) converges to the equilibrium $f_{\infty}$ as $t\rightarrow +\infty$ or not.

Questions concerning convergence to equilibrium

There are several important (and interesting) questions should be answered:

1. (The most natural one) Does the trajectory converge to an equilibrium when $t\rightarrow +\infty$?
2. (Multiple equilibria) It is possible to have multiple equilibria. So the next question is to which equilibrium does the trajectory converge to?
3. (The rate of convergence) How fast is the convergence?

The 1st question is qualitative (one just wants to know that it converges or not) while the 3rd one is quantitative (one wants to know how fast it converges). It should be noted that sometimes the 3rd question turns out to be much more difficult than the 1st one. My favourite example is the imhomogeneous Boltzmann equation: though the convergence to equilibrium was proved around the beginning of twentieth century, it is not until the year 2000 when C. Villani and L. Desvillettes published their result on how fast does it converge to equilibrium.

The method they used is called entropy method, which will be the main tool in this serie of posts.