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

TL;DR

The paper addresses the global well-posedness and relaxation limit for the compressible Euler system with damping in $\mathbb{R}^d$, using a novel hybrid Besov framework that combines low-frequency $L^p$ control with high-frequency $L^2$ control. The authors develop refined product and commutator estimates across multiple frequency bands, including intermediate medium-frequency regimes, to manage nonlinear interactions and extend the allowable range of $p$ to $\infty$. They prove global existence for small hybrid-Besov data and justify the diffusive relaxation limit to the porous medium equation with quantitative convergence rates depending on the data regularity. The methods provide a unified approach to partially dissipative hyperbolic systems and offer potential extensions to hybrid analyses in other models such as compressible Navier–Stokes. The work thus advances understanding of how frequency-split Besov spaces can handle relaxation phenomena in multidimensional hyperbolic–parabolic systems, with implications for stability and long-time behavior in fluid dynamics.

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.

Figures (2)

• Figure 1: Frequency decomposition diagram for $p=6$.
• Figure 2: Frequency Decomposition Diagram for $p<\infty$.

Theorems & Definitions (18)

• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• Lemma 3.2
• Theorem 4.1