We prove the existence of relative finite-energy vanishing viscosity solutions of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, $p(\rho)/\rho = O(1)$ in the limit $\rho \to \infty$. 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.

中文翻译：

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渐近等温气体的可压缩 Navier-Stokes 方程的无粘性极限

我们证明了在渐近等温压力定律的假设下，一维等熵欧拉方程的相对有限能量消失粘度解的存在性，即 $p(\rho)/\rho = O(1)$ in极限 $\rho \to \infty$。该解是作为一维、等熵、可压缩 Navier--Stokes 方程经典解的消失粘度极限获得的。我们的方法依赖于补偿紧凑性的方法在非线性项中严格地传递到极限。我们策略的关键是推导出熵核和相关量的双曲线表示公式；其中，一个特殊的熵对用于在近似解上获得更高的统一可积性估计。依赖于这些表示公式的复杂边界程序然后产生所需的熵耗散措施的紧凑性。反过来，我们证明了由 Navier-Stokes 方程的经典解生成的 Young 测度简化为 Dirac 质量，我们从中推导出所需的收敛到 Euler 方程的解。