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

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 Mathematiques Pures et Appliquees, 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 not equal 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.