Global wellposedness for a class of reaction–advection–anisotropic-diffusion systems

Abstract

We prove existence and uniqueness of global solutions for a class of reaction–advection– anisotropic-diffusion systems whose reaction terms have a “triangular structure”. We thus extend previous results to the case of time–space-dependent anisotropic diffusions and with time–space-dependent advection terms. The corresponding models are in particular relevant for transport processes inside porous media and in situations in which additional migration occurs. The proofs are based on optimal ${L^p}$-maximal regularity results for the general time-dependent linear operator dual to the one involved in the considered systems. As an application, we prove global wellposedness for a prototypical class of chemically reacting systems with mass-action kinetics, involving networks of reactions of the type ${C_{1} + \cdots + C_{P-1} \rightleftharpoons C_{P}}$. Finally, we analyze how a classical a priori ${L^2}$-estimate of the solutions, which holds with this kind of nonlinear reactive terms, extends to our general anisotropic-advection framework. It does extend with the same assumptions for isotropic diffusions and is replaced by an ${L^{(N+1)/N}}$-estimate in the general situation.