Hyperbolic systems with relaxation: Characterization of stiff well-posedness and asymptotic expansions

The Cauchy problem for linear constant-coefficient hyperbolic systems u(t) + Sigma(j) A((j))u(xj) = (1/delta)Bu + Cu in d space dimensions is analyzed. Here (1/delta)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non-trivial null-space. A concept of stiff well-posedness is introduced that ensures solution estimates independent of 0 < delta much less than 1. Stiff well-posedness is characterized algebraically and-under mild assumptions on B-is shown to be equivalent to the existence of a limit of the L-2-solution as delta --> 0. The evolution of the limit is governed by a reduced hyperbolic system, the so-called equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff well-posedness-which is a weaker requirement than the existence of an entropy-leads to the validity of an asymptotic expansion. As an application, we consider a linearized version of a generic model of two-phase now in a porous medium and show stiff well-posedness using a general result on strictly hyperbolic systems. To confirm the theory, the leading terms of the asymptotic expansion are computed and compared with a numerical solution of the full problem. (C) 1999 Academic Press.