Stiff well-posedness for hyperbolic systems with large relaxation terms (linear constant-coefficient problems)

Abstract

The Cauchy problem for linear constant--coefficient hyperbolic systems $u_t+Au_x=(1/\delta)Bu$ is analyzed. Here $(1/\delta)Bu$ is a large relaxation term, and we are mostly interested in the critical case where $B$ has a nontrivial nullspace. A concept of stiff well--posedness is introduced that ensures solution estimates independent of $0<\delta \leq 1$. Under suitable assumptions, we prove convergence of the $L_2$--solution to a limit as $\delta$ tends to zero. The limit solves a reduced strongly hyperbolic system without zero--order term, the so--called equilibrium system, and we present a method to determine this limit system. For 2$\times$2 systems the requirement of stiff well--posedness is shown to be equivalent to the well--known subcharacteristic condition, but in general the subcharacteristic condition is not sufficient for stiff well--posedness. The theory is illustrated by examples.