Vanishing Viscosity Limit of the Compressible Navier--Stokes Equations with General Pressure Law

We prove the convergence of the vanishing viscosity limit of the one-dimensional, isentropic, compressible Navier--Stokes equations to the isentropic Euler equations in the case of a general pressure law. Our strategy relies on the construction of fundamental solutions to the entropy equation that remain controlled for unbounded densities and employs an improved reduction framework to show that measure-valued solutions constrained by the Tartar commutation relation (but with possibly unbounded support) reduce to a Dirac mass. As the Navier--Stokes equations do not admit an invariant region, we work in the finite-energy setting, where a detailed understanding of the high density regime is crucial.