Uniform Asymptotic and Convergence Estimates for the Jin--Xin Model Under the Diffusion Scaling

Roberta Bianchini (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We obtain sharp decay estimates in time in the context of Sobolev spaces, for smooth 4 solutions to the one dimensional Jin-Xin model under the diffusion scaling, which are uniform with 5 respect to the singular parameter of the scaling. This provides convergence to the limit nonlinear 6 parabolic equation both for large time, and for the vanishing singular parameter. The analysis is 7 performed by means of two main ingredients. First, a crucial change of variables highlights the 8 dissipative property of the Jin-Xin system, and allows to observe a faster decay of the dissipative 9 variable with respect to the conservative one, which is essential in order to close the estimates. Next, 10 the analysis relies on a deep investigation on the Green function of the linearized Jin-Xin model, 11 depending on the singular parameter, combined with the Duhamel formula in order to handle the 12 nonlinear terms. 13

Mots clés: asymptotic behavior; dissipation; global existence; decay estimates; diffusive scaling; conservative-dissipative form; BGK models; Green analysis; Relaxation

Domaines: Mathématiques [math]

Roberta Bianchini. Uniform Asymptotic and Convergence Estimates for the Jin--Xin Model Under the Diffusion Scaling. SIAM Journal on Mathematical Analysis, 2018, 50 (2), pp.1877-1899. ⟨hal-02012705⟩