Category Archives: Entropy

About Entropy method for PDEs, especially Reaction diffusion systems.

Keywords: Entropy, conversation laws, Lyapunov functional, Reaction diffusion systems.

Entropy method for chemical reaction-diffusion systems II: A linear system

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 … Continue reading

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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 … Continue reading

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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. … Continue reading

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Exponential convergence to equilibrium for a class of chemical reactions

In this post, we will see how a Lyapunov functional will help us to prove the explicit convergence to equilibrium for some chemical reactions. We consider a single reversible reaction with the forward and backward reaction rates are assumed to … Continue reading

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Well posedness of a coupled PDE-ODE system

In this post, I simply give a proof of the well posedness of a “simple” coupled PDE-ODE system (which we mentioned in this post). The idea is based on “localized method”. Let is a bounded set with smooth boundary . … Continue reading

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Control the boundary value

Consider the problem                  (1) in an open, convex, bounded domain with the boundary condition on and intial data . We assume that . By maximum principle, we know that if is a … Continue reading

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The Csiszár-Kullback inequality

Today we give an inequality, which is very useful in proving the convergence of a solution to PDEs tends to the stationary solution as time tends to infinity. Lemma. Let  be a domain and let satisfy and . Furthermore, let … Continue reading

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