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A Glimm type functional for a special Jin–Xin relaxation model

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  • Medicine


PII: S0294-1449(00)00124-4 Ann. Inst. Henri Poincaré Anal. nonlinear 18, 1 (2001) 19–42 Ó 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0294-1449(00)00124-4/FLA A GLIMM TYPE FUNCTIONAL FOR A SPECIAL JIN–XIN RELAXATION MODEL Stefano BIANCHINI 1 SISSA-ISAS, via Beirut 2-4, 34014 Trieste, Italy Manuscript received 18 January 2000 ABSTRACT. – We consider a special case of the Jin–Xin relaxation systems ut + vx = 0, vt + λ2ux = (F (u)− v)/ε. (∗) We assume that the integral curves of the eigenvectors ri of DF(u) are straight lines. In this setting we prove that for every initial data u,v with sufficiently small total variation the solution (uε, vε) of (∗) is well defined for all t > 0, and its total variation satisfies a uniform bound, independent of t, ε. Moreover, as ε tends to 0+, the solutions (uε, vε) converge to a unique limit (u(t), v(t)): u(t) is the unique entropic solution of the corresponding hyperbolic system ut + F(u)x = 0 and v(t, x)= F(u(t, x)) for all t > 0, a.e. x ∈R. The proofs rely on the introduction of a new functional for the solutions of (∗), corresponding to the Glimm interaction potential for the approaching waves of different families. Ó 2001 Éditions scientifiques et médicales Elsevier SAS AMS classification: 35L65 RÉSUMÉ. – Nous considérons un cas special des systèmes de relaxation ut + vx = 0, vt + λ2ux = (F (u)− v)/ε. (∗) Nous supposons que les courbes intégrales des vecteurs propres ri de DF(u) sont des droites. Sous ces hypothèses, nous prouvons que pour chaques données initiales u,v avec une variation totale suffisamment petite la solution (uε, vε) de (∗) est bien définie pour tout t > 0, et sa variation totale satisfait une borne uniforme, indépendante de t, ε. De plus, quand ε tend vers 0+, les solutions (uε, vε) convergent vers une unique limite (u(t), v(t)) : u(t) est l’unique solution entropique du système hyperbolique correspondant ut + F(u)x = 0 et v(t, x)= F(u(t, x)) pour tout t > 0, p.p. x ∈ R. Les preuv

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