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

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  • 3 February 2026
    • preprint
    • Published by HAL in HAL
Abstract
We investigate the global well-posedness of the compressible Euler system with damping in R^d (d ≥ 1) 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 lowfrequency components are controlled in L p -based norms with p ∈ [2, max{4, 2d/(d -2)}]. Motivated by the observation that the limit system is well-posed in Lp -based spaces for p ∈ [2, ∞), 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.

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