Stabilité des profils de chocs dans les systèmes de lois de conservation

Abstract : We are interested here in the theoretical study of the stability of shock profiles for several approximations of hyperbolic monodimensional systems of conservation laws. We consider in the first part the continuous profiles for the semi-linear approximation and for equations with diffusive and dispersive effects. We obtain necessary conditions of spectral stability using Evans function techniques and more precisely the Gap Lemma that was proved by Gardner and Zumbrun. For the semi-linear relaxation, we illustrate the necessity of the condition for a Lax 2-shock of a system of two conservation laws by computing an unstable shock profile, that we simulate numerically through a splitting scheme. We also show that, when the relaxation speed tends to infinity, the limit of the Evans function that is associated to the semi-linear relaxation approximation is the Evans function associated to a scalar viscous approximation. The second part is devoted to discrete stationary shock profiles. We obtain a necessary condition of spectral stability for the Lax-Wendroff scheme by adapting the techniques we used in the continuous case. At last, we study the Green's function associated with the modified Lax-Friedrichs scheme and we get estimates that are similar to the ones Zumbrun and Howard obtained for the viscous approximation.
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Pauline Lafitte-Godillon. Stabilité des profils de chocs dans les systèmes de lois de conservation. Mathématiques [math]. Ecole normale supérieure de lyon - ENS LYON, 2001. Français. ⟨tel-00396376⟩

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