Compressible Euler equations with time-dependent damping in the critical regularity setting: global well-posedness and strong relaxation limit
Abstract
We investigate the relaxation problem and the diffusion phenomenon for the compressible Euler system with a time-dependent damping coefficient of the form µ (1+t) λ in R d (d ≥ 1). We establish uniform regularity estimates with respect to the relaxation parameter ε and prove the global wellposedness of classical solutions to the Cauchy problem. In addition, we justify the global-in-time strong convergence of the solutions towards those of a general porous medium-type diffusion system, with an explicit rate of convergence, and for ill-prepared initial data. The core of our proof relies on a refined hypocoercivity framework combined with a new time-dependent frequency decomposition, both adapted to handle damping terms with time-dependent coefficients. This enables us to treat the overdamped regime λ ∈ (-∞, 0) and the underdamped regime λ ∈ (0, 1) for any µ > 0, and also the borderline critical case λ = 1 under the improved condition µ > 2ε 2 .
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BibTeX
@misc{crinbarat2026compressible,
title = {{Compressible Euler equations with time-dependent damping in the critical regularity setting: global well-posedness and strong relaxation limit}},
author = {Timothée Crin-Barat and Xinghong Pan and Ling-Yun Shou and Qimeng Zhu},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05404720v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compressible Euler equations with time-dependent damping in the critical regularity setting: global well-posedness and strong relaxation limit
AU - Timothée Crin-Barat
AU - Xinghong Pan
AU - Ling-Yun Shou
AU - Qimeng Zhu
PY - 2026
DA - 2026/09/07
PB - HAL
UR - https://hal.science/hal-05404720v2
AB - We investigate the relaxation problem and the diffusion phenomenon for the compressible Euler system with a time-dependent damping coefficient of the form µ (1+t) λ in R d (d ≥ 1). We establish uniform regularity estimates with respect to the relaxation parameter ε and prove the global wellposedness of classical solutions to the Cauchy problem. In addition, we justify the global-in-time strong convergence of the solutions towards those of a general porous medium-type diffusion system, with an explicit rate of convergence, and for ill-prepared initial data. The core of our proof relies on a refined hypocoercivity framework combined with a new time-dependent frequency decomposition, both adapted to handle damping terms with time-dependent coefficients. This enables us to treat the overdamped regime λ ∈ (-∞, 0) and the underdamped regime λ ∈ (0, 1) for any µ > 0, and also the borderline critical case λ = 1 under the improved condition µ > 2ε 2 .
ER -