Abstract:We consider a special case of the Jin--Xin relaxation systemsu_t + v_x = 0, \qquad v_t + \lambda^2 u_x = (F(u) - v) /\epsilon. \eqno(*)We assume that the integral curves of the eigenvectors $r_i$ of $DF(u)$ are straight lines.In his setting we prove that for every initial data $u, v$ with sufficiently small total variation the solution $(u^{\epsilon}, v^{\epsilon})$ of $(*)$ is well defined for all $t > 0$, and its total variation satisfies a uniform bound, independent of $t, \epsilon$. Moreover, as $\epsilon$ tends to $0^+$, the solutions $(u^\epsilon, v^\epsilon)$ converge to a unique limit $(u(t), v(t))$: $u(t)$ is the unique entropic solution of the corresponding hyperbolic system $u_t + F(u)_x = 0$ and $v(t,x) = F(u(t,x))$ for all $t > 0$, a.e. $x \in \R$.

The proofs rely on the introduction of a new functional for the solutions of $(*)$, corresponding to the Glimm potential for the approaching waves of different families.

**Paper:**- Available as PostScript.
**Author(s):**- Stefano Bianchini, <bianchin@sissa.it>
**Publishing information:****Comments:****Submitted by:**- <bianchin@sissa.it> November 26 1999.

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