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首页> 外文期刊>SIAM Journal on Mathematical Analysis >Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations
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Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations

机译：一维可压缩Navier-Stokes方程整体强解的存在性和唯一性

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摘要

We consider Navier-Stokes equations for compressible viscous fluids in one dimension. It is a well-known fact that if the initial datum are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists locally in time. In this paper, we show that under the same hypothesis, the density remains bounded by below by a positive constant uniformly in time, and that strong solutions therefore exist globally in time. Moreover, while most existence results are obtained for positive viscosity coefficients, the present result holds even if the viscosity coefficient vanishes with the density. Finally, we prove that the solution is unique in the class of weak solutions satisfying the usual entropy inequality. The key point of the paper is a new entropy-like inequality introduced by Bresch and Desjardins for the shallow water system of equations. This inequality gives additional regularity for the density (provided such regularity exists at initial time).
机译：我们考虑一维可压缩粘性流体的Navier-Stokes方程。众所周知的事实是，如果初始基准面是光滑的，并且初始密度在下方受到正常数的限制，那么及时就会在局部存在强解。在本文中，我们表明，在相同的假设下，密度在时间上始终保持均匀地由下面的正常数所包围，因此强的解决方案因此在时间上全局存在。此外，尽管对于正粘度系数可获得大多数存在结果，但是即使粘度系数随密度消失，该结果仍然成立。最后，我们证明了该解在满足通常的熵不等式的弱解类中是唯一的。本文的重点是Bresch和Desjardins针对浅水方程组引入的新的类似熵的不等式。这种不等式为密度提供了其他规律性（前提是这种规律性在初始时间就存在）。

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