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We here investigate a modification of the compressible barotropic Eulersystem with friction, involving a fuzzy nonlocal pressure term in place of theconventional one. This nonlocal term is parameterized by $\epsilon$ > 0 andformally tends to the classical pressure when $\epsilon$ approaches zero. Thecentral challenge is to establish that this system is a reliable approximationof the classical compressible Euler system. We establish the global existenceand uniqueness of regular solutions in the neighborhood of the static statewith density 1 and null velocity. Our results are demonstrated independently ofthe parameter $\epsilon$, which enable us to prove the convergence of solutionsto those of the classical Euler system. Another consequence is the rigorousjustification of the convergence of the mass equation to various versions ofthe porous media equation in the asymptotic limit where the friction tends toinfinity. Note that our results are demonstrated in the whole space, whichnecessitates the use of a suitable L^1-in-time space framework.

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