### Hyperbolic limit of the Jin–Xin relaxation model

Stefano Bianchini

Abstract: We consider the special Jin–Xin relaxation model

ut+A(u)ux = ε(uxxutt).

We assume that the initial data (u0, εu0,t) are sufficiently smooth and close to (u, 0) in L and have small total variation. Then we prove that there exists a solution (uε(t), εuεt(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 uε(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

ut+A(u)ux = 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}.

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Author(s):
Stefano Bianchini, <bianchini@iac.cnr.it>
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