In the last post, I’ve shown you how does the entropy method work in the case of a single equation: heat equation with homogeneous Neumann boundary condition. The application of entropy method in systems is more involved and needs more efforts to bring out what we want.

In this post, we first try to apply the method to a *linear system. *The application to nonlinear systems is more complicated and will be revealed in next posts.

**Modelling the reaction-diffusion system**

Assuming that a reaction

takes place in a bounded domain with the forward- and backward-reaction constant rates are normalised to . The corresponding linear RD system for two unknown functions and reads as

and initial data and .

**Normalised volume and average value**

Throughout this post, we will assume that has a normalised volume, that is . For a function , we denote by

the average value of .

**Conservation of mass**

The solution to this solution admits one *conservation of mass *

for all

**Constant equilibrium**

The unique equilibrium is .

**Convergence to equilibrium?**

To prove the convergence to equilibrium as we consider the *quadratic entropy functional *

,

where is the usual norm, and its *entropy dissipation*

.

We now aim to prove an* entropy-entropy dissipation estimate* of the form

for some .

To prove this inequality (*) we follow the following scheme, which is later turned out to be very effective to apply to nonlinear systems,

*Step 0: (Decompose the difference of entropy)*

Using the mass conservation and values of the equilibrium we can show that

where

and

.

In the next steps, we will try to control and separately.

*Step 1: (Role of diffusion)*

To control , we use the Poincaré inequaltiy to have

and .

Hence, we have

.

*Step 2: (Reaction of averages)*

Applying the Jensen’s inequality (see here for more general inequalities) we have

where we used the mass conservation at the last step. Therefore, we have

.

*Step 3: (Combining steps 1 and 2)*

From steps 1 and 2, we easily see that

with , thus, by the Gronwall inequality

.

On the other hand, by the mass conservation we easily see that

.

In conclusion, we have proved

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## About baotangquoc

Lecturer
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
No 1, Dai Co Viet Street, Hanoi