Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: application to the nonlinearly damped p-system

TL;DR

This paper introduces a Fourier-free, physical space approach to analyze the long-time decay of solutions for partially dissipative hyperbolic systems, including nonlinear models, under sharp stability conditions.

Contribution

It develops a new physical space hyperbolic hypocoercivity method that avoids Fourier analysis, enabling sharper decay estimates for nonlinear hyperbolic systems.

Findings

Solutions decay exponentially at high frequencies and polynomially at low frequencies.

Established new decay estimates for initial data in weighted Sobolev spaces.

Proved logarithmic stability for the nonlinear damped p-system.

Abstract

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform that we avoid here, developing the "physical space version" of the hyperbolic hypocoercivity approach introduced by Beauchard and Zuazua, to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows to recover the natural heat-like time-decay of solutions under sharp rank conditions, without employing Fourier analysis or $L^1$ assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped $p$-system.