Hyperbolic Systems with Relaxation: Characterization of Stiff Well-Posedness and Asymptotic Expansions

The Cauchy problem for linear constant-coefficient hyperbolic systems utq j A j.uxjs 1rd.BuqCu in d space dimensions is analyzed. Here 1rd.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-d<1. Stiff well-posedness is characterized algebraically and}under mild assumptions on B}is shown to be equi¨alent to the existence of a limit of the L2-solution as dª0. 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 flow 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