2021/05/18 by Crin-Barat, Timothée, Danchin, Raphaël · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2105.08333
We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole space ℝd with d≥2. Following our recent work [10] dedicated to the one-dimensional case, we establish the existence of global strong solutions and decay estimates in the critical regularity setting whenever the system under consideration satisfies the so-called (SK) (for Shizuta-Kawashima) condition. Our results in particular apply to the compressible Euler system with damping in the velocity equation. Compared to the papers by Kawashima and Xu [27, 28] devoted to similar issues, our use of hybrid Besov norms with different regularity exponents in low and high frequency enable us to pinpoint optimal smallness conditions for global well-posedness and to get more accurate information on the qualitative properties of the constructed solutions. A great part of our analysis relies on the study of a Lyapunov functional in the spirit of that of Beauchard and Zuazua in [2]. Exhibiting a damped mode with faster time decay than the whole solution also plays a key role.