The entropy dissipation method for spatially inhomogeneous reaction-diffusion-type systems
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AbstractWe study the long-time asymptotics of reaction-diffusion-type systems that feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimizing) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so-called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion-convection equations, and the main goal of this paper is to study its generalization to systems of partial differential equations that contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of a reaction-diffusion-convection system arising in solid-state physics as a paradigm for general nonlinear systems. © 2008 The Royal Society.
CitationDi Francesco M, Fellner K, Markowich PA (2008) The entropy dissipation method for spatially inhomogeneous reaction-diffusion-type systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464: 3273–3300. Available: http://dx.doi.org/10.1098/rspa.2008.0214.
SponsorsThis article was supported by the KAUST Investigator Award 2008 and the Wolfson Research Merit Award (Royal Society) of P. A. M. M. D. F. was partially supported by the Italian research project ' Modelli iperbolici non lineari in fluido dinamica' (INdAM GNAMPA 2008).
PublisherThe Royal Society