TY - JOUR

T1 - Inviscid limit of the compressible Navier–Stokes equations for asymptotically isothermal gases

AU - Schrecker, Matthew R.i.

AU - Schulz, Simon

PY - 2020/11/1

Y1 - 2020/11/1

N2 - We prove the existence of a relative finite-energy solution of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, p (p)/p = 0(1) in the limit p →∞. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier–Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier–Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.

AB - We prove the existence of a relative finite-energy solution of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, p (p)/p = 0(1) in the limit p →∞. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier–Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier–Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.

U2 - 10.1016/j.jde.2020.06.018

DO - 10.1016/j.jde.2020.06.018

M3 - Article

SN - 0022-0396

VL - 269

SP - 8640

EP - 8685

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 10

ER -