The compressible Euler system with nonlocal pressure: global existence and relaxation

We here investigate a modification of the compressible barotropic Euler system with friction, involving a fuzzy nonlocal pressure term in place of the conventional one. This nonlocal term is parameterized by \(\varepsilon > 0\) and formally tends to the classical pressure when \(\varepsilon \) approaches zero. The central challenge is to establish that this system is a reliable approximation of the classical compressible Euler system. We establish the global existence and uniqueness of regular solutions in the neighborhood of the static state with density 1 and null velocity. Our results are demonstrated independently of the parameter \(\varepsilon ,\) which enable us to prove the convergence of solutions to those of the classical Euler system. Another consequence is the rigorous justification of the convergence of the mass equation to various versions of the porous media equation in the asymptotic limit where the friction tends to infinity. Note that our results are demonstrated in the whole space, which necessitates to use the \(L^1(\mathbb {R}_+; \dot{B}^\sigma _{2,1}(\mathbb {R}^d))\) spaces framework.