Convergence of Singular Limits for Multi-D Semilinear Hyperbolic Systems to Parabolic Systems

TL;DR

This paper studies the zero-relaxation limit of multi-dimensional semilinear hyperbolic systems, demonstrating their convergence to parabolic systems through energy estimates and compensated compactness, with applications to reaction-diffusion models.

Contribution

It introduces algebraic structure conditions and dissipativity assumptions that enable rigorous analysis of the singular limit from hyperbolic to parabolic systems.

Findings

Established uniform energy estimates in epsilon

Proved convergence to parabolic systems using compensated compactness

Provided examples applying the theory to reaction-diffusion systems

Abstract

In this paper we investigate the zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W_{t}(x,t) + (1/epsilon) A(x,D) W(x,t) = (1/epsilon^2) B(x,W(x,t)) + (1/epsilon) D(W(x,t)) + E(W(x,t)). We analyse the singular convergence, as epsilon tends to 0, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce ``energy estimates'', uniformly in epsilon, by assuming the existence of a symmetrizer having the so called block structure and by assuming ``dissipativity conditions'' on B. (iii) We perform the convergence analysis by using generalizations of Compensated Compactness due to Tartar and Gerard. Finally we include examples which show how to use our theory to approximate prescribed general quasilinear parabolic systems, satisfying Petrowski parabolicity condition, or general reaction diffusion systems.