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
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
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
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