WIAS Preprint No. 2349, (2016)

Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities


  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Mittnenzweig, Markus
    ORCID: 0000-0002-8502-1702

2010 Mathematics Subject Classification

  • 35K57 35B40 35Q79 92E20


  • Energy-reaction-diffusion systems, vector-valued inequalities, cross diffusion, log-Sobolev inequality, entropy functional, exponential decay of relative entropy, convexity method




We discuss how the recently developed energy-dissipation methods for reactiondi usion 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.

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