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In this paper we investigate the zero-relaxation limit of the followingmulti-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 whichleads to a limit system of parabolic type. The analysis is carried out by usingthe following steps:  (i) We single out algebraic ``structure conditions'' on the full system,motivated by formal asymptotics, by some examples of discrete velocity modelsin kinetic theories.  (ii) We deduce ``energy estimates'', uniformly in epsilon, by assuming theexistence 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 ofCompensated Compactness due to Tartar and Gerard.  Finally we include examples which show how to use our theory to approximateprescribed general quasilinear parabolic systems, satisfying Petrowskiparabolicity condition, or general reaction diffusion systems.

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