Long-time behavior of solutions to the M1 model with boundray effect

TL;DR

This paper studies the long-term behavior of solutions to the M1 model with boundary effects, showing they converge to a nonlinear parabolic equation with improved rates using energy estimates and Green's functions.

Contribution

It introduces a generalized system combining the M1 model with damped Euler equations and proves global existence and convergence to Darcy's law with enhanced rates.

Findings

Solutions exist globally and tend to a nonlinear parabolic equation

Improved convergence rates over previous results

Effective use of energy estimates and Green's function methods

Abstract

In this paper, we are concerned with the asymptotic behavior of solutions of M1 model on quadrant. From this model, combined with damped compressible Euler equations, a more general system is introduced. We show that the solutions to the initial boundary value problem of this system globally exist and tend time-asymptotically to the corresponding nonlinear parabolic equation governed by the related Darcy's law. Compared with previous results on compressible Euler equations with damping obtained by Nishihara and Yang in [24], and Marcati, Mei and Rubino in [16], the better convergence rates are obtained. The approach adopted is based on the technical time-weighted energy estimates together with the Green's function method.