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 [10] 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 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 in particular applies to the multi-dimensional isentropic compressible Euler system with relaxation, and provide us with bounds that are independent of the relaxation parameter. As a consequence, we justify rigorously the relaxation limit to the porous media equation and exhibit explicit rates of convergence for suitable norms, a completely new result to the best of our knowledge.