Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling

TL;DR

This paper establishes uniform decay estimates for the one-dimensional Jin-Xin model under diffusion scaling, demonstrating convergence to a nonlinear parabolic limit while analyzing the Green function and dissipative properties.

Contribution

It provides sharp, uniform decay estimates for the Jin-Xin model under diffusion scaling, revealing convergence to the parabolic limit with novel analysis techniques.

Findings

Uniform decay estimates in Sobolev spaces

Convergence to nonlinear parabolic equation

Analysis of Green function depending on the singular parameter

Abstract

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