The compressible Euler system with damping in hybrid Besov spaces: global well-posedness and relaxation limit

TL;DR

This paper proves the global well-posedness of the compressible Euler system with damping in hybrid Besov spaces and analyzes its relaxation limit to the porous medium equation, extending previous results to a broader range of function spaces.

Contribution

The authors extend the low-frequency analysis to the full range p∈[2,∞) in hybrid Besov spaces, providing a more unified framework for the relaxation limit of the Euler system.

Findings

Established refined product and commutator estimates.

Proved global well-posedness in hybrid Besov spaces.

Analyzed the relaxation limit to the porous medium equation.

Abstract

We investigate the global well-posedness of the compressible Euler system with damping in Rd (d\geq1) and its relaxation limit toward the porous medium equation. In [12], the first author and Danchin studied these two problems in hybrid Besov spaces, where the high-frequency components of the solution are bounded in L2-based norms, while the low-frequency components are controlled in Lp-based norms with p\in[2,\max{4,\frac{2d}{d-2}}]. Motivated by the observation that the limit system is well-posed in Lp-based spaces for p\in[2, \infty), we extend the low-frequency analysis to this full range, thereby providing a more unified framework for studying such relaxation limits. The core of our proof consists in establishing refined product and commutator estimates describing sharply the interactions between the high, medium, and low-frequency regimes. A key observation underlying our analysis is that the product of two functions localized at low frequencies generates only interactions between low and medium frequencies, never purely high-frequency ones. Consequently, for a suitable choice of frequency threshold, the high-frequency projection of the product of two functions localized low frequencies vanishes.