Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems

Abstract:

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: \[ W_{t}(x,t) + \frac {1}{\varepsilon }A(x,D)W(x,t)= \frac {1}{\varepsilon ^2} B(x,W(x,t))+\frac {1}{\varepsilon } D(W(x,t))+E(W(x,t)).\] We analyze the singular convergence, as $\varepsilon \downarrow 0$, 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 $\varepsilon$, by assuming the existence of a symmetrizer having the so-called block structure and by assuming “dissipativity conditions” on $B$.

[(iii)] We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard.

Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.