Preprint 2004-060

Hyperbolic limit of the Jin–Xin relaxation model

Abstract: We consider the special Jin–Xin relaxation model

u_t+A(u)u_x = ε(u_xx−u_tt).

We assume that the initial data (u₀, εu_0,t) are sufficiently smooth and close to (u, 0) in L^∞ and have small total variation. Then we prove that there exists a solution (u^ε(t), εu^ε_t(t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz continuously in the L¹ norm w.r.t. the initial data and time.

We then take the limit ε→0, and show that u^ε(t) tends to a unique Lipschitz continuous semigroup \mathcal{S} on a domain \mathcal{D} containing the functions with small total variation and close to u. The semigroup \mathcal{S} defines a semigroup of relaxation limiting solutions} to the quasilinear non conservative system

u_t+A(u)u_x = 0.

Moreover this semigroup coincides with the trajectory of a Riemann Semigroup, which is determined by the unique Riemann solver compatible with \eqref{E:speJX01}.

Paper:

Available as PDF (400 Kbytes), Postscript (2 Mbytes) or gzipped PostScript (496 Kbytes; uncompress using gunzip).

Author(s):

Stefano Bianchini, <bianchini@iac.cnr.it>

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Submitted by:

<bianchini@iac.cnr.it> November 5 2004.