论文标题
在常规解决方案上，用于具有退化粘度和远场真空的三维完全可压缩的Navier-Stokes方程
On regular solutions for three-dimensional full compressible Navier-Stokes equations with degenerate viscosities and far field vacuum
论文作者
论文摘要
在本文中，考虑了三维（3-D）完整可压缩的Navier-Stokes方程（CNS）的CAUCHY问题。首先，当剪切和散装粘度系数都取决于Chapman-Enskog的电力定律中的绝对温度$θ$（$θ^ν$，带有$ν> 0 $），基于对该系统内在的奇异结构的一些详尽的分析，我们确定了一类PINITIC PATICT $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $的$ $ $ $ $ f。 $ S $。此外，还表明，在这种常规解决方案的寿命中，速度保持在不均匀的sobolev空间中，即$ u \ in h^3（\ mathbb {r}^3）$，$ s $，$ s $在整个空间和整个质量，整个能源的范围内都具有统一的有限的下限和上限，总体能量和总体能源是总体上满意的。请注意，由于真空的出现，动量方程在时间演化和粘性应力张量中都退化，而多变量气体的物理熵的表现也很单一，这使得对相应的适合良好性的研究具有挑战性。为了证明存在，我们首先通过考虑一些新变量来引入一个扩大的重新重新制定结构，这些变量可以将完整中枢神经系统的变性转移到与$ S $相关的一些特殊源术语的可能奇异性上，然后进行一些针对此重新计算系统精心设计的奇异加权能量估计。
In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier-Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute temperature $θ$ in a power law ($θ^ν$ with $ν>0$) of Chapman-Enskog, based on some elaborate analysis of this system's intrinsic singular structures, we identify one class of initial data admitting a local-in-time regular solution with far field vacuum in terms of the mass density $ρ$, velocity $u$ and entropy $S$. Furthermore, it is shown that within its life span of such a regular solution, the velocity stays in an inhomogeneous Sobolev space, i.e., $u\in H^3(\mathbb{R}^3)$, $S$ has uniformly finite lower and upper bounds in the whole space, and the laws of conservation of total mass, momentum and total energy are all satisfied. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, and the physical entropy for polytropic gases behaves singularly, which make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the full CNS to the possible singularities of some special source terms related with $S$, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.
