Preprint 1997-032

Stiff Well-Posedness and Asymptotic Expansions for Hyperbolic Systems with Relaxation

Abstract: The Cauchy problem for linear constant--coefficient hyperbolic systems $u_t+ \sum_{j} \Aj u_\xj = (1/\delta)Bu+Cu$ 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 $0< \delta \ll 1$. Under suitable assumptions, an asymptotic expansion for small $\delta$ is presented. Furthermore, we give sharp conditions for the existence of a limit of the $L_2$--solution as $\delta$ tends to zero. The evolution of this limit is governed by a reduced hyperbolic system, the so--called equilibrium system. The original system and the equilibrium system are related by a phase speed condition.

The theory is applied to a linearized version of a generic model of two--phase flow in a porous medium. Stiff well--posedness of this system is analyzed in detail, and the leading terms of the expansion are computed numerically.

Paper:

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Title:

Stiff well-posedness and asymptotic expansions for hyperbolic systems with relaxation

Author(s):

Jens Lorenz, <lorenz@math.unm.edu>

Hans Joachim Schroll, <schroll@igpm.rwth-aachen.de>

Publishing information:

Institut Mittag--Leffler, Report No. 3, 1997/98

Comments:

Submitted by:

<schroll@igpm.rwth-aachen.de> November 18 1997.