Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped p-system
Crin-Barat T, Shou LY, Zuazua E (2025)
Publication Language: English
Publication Type: Journal article
Publication year: 2025
Journal
Book Volume: 42
Pages Range: 1165-1218
Journal Issue: 5
URI: https://arxiv.org/abs/2308.08280
DOI: 10.4171/AIHPC/128
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 us 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, which we avoid here, developing the “physical space version” of the hyperbolic hypocoercivity approach introduced in Beauchard and Zuazua [Arch. Ration. Mech. Anal. 199 (2011), 177–227], to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows us to recover the natural heat-like time decay of solutions under sharp rank conditions, without employing Fourier analysis or L1 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.
Authors with CRIS profile
Timothée Crin-Barat
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
China (CN)How to cite
APA:
Crin-Barat, T., Shou, L.Y., & Zuazua, E. (2025). Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped p-system. Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 42(5), 1165-1218. https://doi.org/10.4171/AIHPC/128
MLA:
Crin-Barat, Timothée, Ling Yun Shou, and Enrique Zuazua. "Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped p-system." Annales de l'Institut Henri Poincaré - Analyse Non Linéaire 42.5 (2025): 1165-1218.
BibTeX: Download