Global existence for partially dissipative hyperbolic systems in the Lp framework, and relaxation limit

• Published in: Mathematische annalen Vol. 386; no. 3-4; pp. 2159 - 2206
• Main Authors: Crin-Barat, Timothée, Danchin, Raphaël
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2023; Springer Nature B.V; Springer Verlag
• Subjects: Analysis of PDEs; Compressibility; Dissipation; Existence theorems; Function space; Hyperbolic systems; Mathematics; Mathematics and Statistics; Norms; Porous media; Well posed problems; 35Q35; 76N10; 1991 Mathematics Subject Classification. 35Q35; relaxation limit; critical regularity; 76N10 Hyperbolic systems; partially dissipative
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-022-02450-4

Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper (Crin-Barat and Danchin in Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case . Published online in Journal de Mathématiques Pures et Appliquées, 2022) to a functional framework where the low frequencies of the solution are only bounded in L p -type spaces with p larger than 2. This unusual setting is in sharp contrast with the non-dissipative case (even linear), where well-posedness in L p for p ≠ 2 fails (Brenner in Math Scand 19:27–37, 1966). Our new framework enables us to prescribe weaker smallness conditions for global well-posedness and to get a more accurate information on the qualitative properties of the constructed solutions. Our existence theorem applies to the multi-dimensional isentropic compressible Euler system with relaxation, and provide us with bounds that are independent of the relaxation parameter for general ill-prepared data, provided they are small enough. As a consequence, we justify rigorously the relaxation limit to the porous media equation and exhibit explicit rates of convergence in suitable norms, a completely new result to the best of our knowledge.