论文标题
低粘性可压缩的Navier-Stokes方程的非唯一性
Non-uniqueness for the hypo-viscous compressible Navier-Stokes equations
论文作者
论文摘要
我们在所有维度的一般压力法律下研究等等屈光度低粘性的Navier-Stokes方程（CNS）的Cauchy问题。对于所有$α\ in（0,1）$ $α\ $α$的所有低视率$（ - δ）^α$，我们证明存在具有相同初始数据的无限弱解决方案。这提供了弱解决粘性压缩液的第一个非唯一性结果。我们的证明具有密度和动量的构建块的新结构，这些结构尊重可压缩结构。它还适用于可压缩的Euler方程和不可压缩的Navier-Stokes方程（INS）。特别是，鉴于Ladyženskaja-prodi-serrin标准，获得的$ l^2_tc_x $ nodientions of viscous ins viscous Ins的解决方案很敏锐，并且揭示了$ lavise the $ l^2_tc_tc_x $ a = 1 $ a = 1 $是尖锐的粘度阈值。此外，我们证明，可以将Hölder连续的弱解对可压缩的Euler方程获得，作为一系列弱溶液序列的强烈消失的粘度极限。
We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations (CNS) under general pressure laws in all dimensions $d\geq 2$. For all hypo-viscosities $(-Δ)^α$ with $α\in (0,1)$, we prove that there exist infinitely many weak solutions with the same initial data. This provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Our proof features new constructions of building blocks for both the density and momentum, which respect the compressible structure. It also applies to the compressible Euler equations and the hypo-viscous incompressible Navier-Stokes equations (INS). In particular, in view of the Ladyženskaja-Prodi-Serrin criteria, the obtained non-uniqueness of $L^2_tC_x$ weak solutions to the hypo-viscous INS is sharp, and reveals that $α=1$ is the sharp viscosity threshold for the well-posedness in $L^2_tC_x$. Furthermore, we prove that the Hölder continuous weak solutions to the compressible Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of weak solutions to the hypo-viscous CNS.
