Singular limits for nonlinear hyperbolic systems

The paper is concerned with singular limits for the nonlinear hyperbolic system Wt+A(x,W,D)W=1εF(W),(1) where W=W(x,t) takes values in RN, x∈Rd, A(x,W,D) is a first-order differential or pseudodifferential operator, F(W) denotes the nonhomogeneous term, and ε>0 represents a relaxation parameter. The focus is on convergence as ε→0. The authors begin with the fast-flow scales asymptotic expansions by following the approach of C. Lattanzio and W.-A. Yong [Comm. Partial Differential Equations 26 (2001), no. 5-6, 939--964; MR1843290 (2002k:35029)] in the quasilinear case of (1), then show some examples of limits arising in physics problems. Then the authors follow the general framework proposed by P. Marcati and B. Rubino [J. Differential Equations 162 (2000), no. 2, 359--399; MR1751710 (2001d:35125)] to study a 2×2 nonlinear hyperbolic systems in the 1-D case. Finally, the last section is devoted to the mathematical theory of semilinear hyperbolic systems, with nonconstant coefficients in the multi-dimensional case.