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Mielke, Alexander; Mittnenzweig, Markus

Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities. (English) Zbl 1390.35156

J. Nonlinear Sci. 28, No. 2, 765-806 (2018).
Summary: We discuss how the recently developed energy dissipation methods for reaction diffusion systems can be generalized to the non-isothermal case. For this, we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions, we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

Cited in 9 Documents

MSC:

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35Q79 PDEs in connection with classical thermodynamics and heat transfer
92E20 Classical flows, reactions, etc. in chemistry

Keywords:

energy-reaction-diffusion systems; vector-valued inequalities; cross diffusion; log-Sobolev inequality; entropy functional; exponential decay of relative entropy; convexity method

Software:

GENERIC
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References:

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