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We establish the vanishing viscosity limit of the Navier-Stokes equations tothe isentropic Euler equations for one-dimensional compressible fluid flow. Forthe Navier-Stokes equations, there exist no natural invariant regions for theequations with the real physical viscosity term so that the uniform sup-norm ofsolutions with respect to the physical viscosity coefficient may not bedirectly controllable and, furthermore, convex entropy-entropy flux pairs maynot produce signed entropy dissipation measures. To overcome thesedifficulties, we first develop uniform energy-type estimates with respect tothe viscosity coefficient for the solutions of the Navier-Stokes equations andestablish the existence of measure-valued solutions of the isentropic Eulerequations generated by the Navier-Stokes equations. Based on the uniformenergy-type estimates and the features of the isentropic Euler equations, weestablish that the entropy dissipation measures of the solutions of theNavier-Stokes equations for weak entropy-entropy flux pairs, generated bycompactly supported $C^2$ test functions, are confined in a compact set in$H^{-1}$, which lead to the existence of measure-valued solutions that areconfined by the Tartar-Murat commutator relation. A careful characterization ofthe unbounded support of the measure-valued solution confined by the commutatorrelation yields the reduction of the measure-valued solution to a Delta mass,which leads to the convergence of solutions of the Navier-Stokes equations to afinite-energy entropy solution of the isentropic Euler equations.

Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow