论文标题
全局球形对称的解决方案，可压缩可压缩的Navier-Stokes方程，并具有大数据和远场真空
Global spherically symmetric solutions to degenerate compressible Navier-Stokes equations with large data and far field vacuum
论文作者
论文摘要
我们考虑在$ \ Mathbb r^d $ $（d = 2 = 2 \ \ \ \ \ \ \ \ \\ text {or} \ 3）$中，我们考虑了等粒子可压缩Navier-Stokes方程（\ textbf {cns}）的初始边界值问题（IBVP）。根据对该系统的非线性结构的某些分析，将粘度系数作为质量密度$ρ$的常数倍数，我们证明了（大）具有球形对称性的（大）初始数据的唯一球形对称的经典解决方案的全球存在，并在某些不均匀的Sobolev空间中使用球形对称性和远处的场真空。此外，我们获得的解决方案具有保守的总质量和有限的总能量。 $ρ$在考虑的域中保持阳性，但在远场中降至零，这与总质量是保守的事实是一致的，\ textbf {cns}是一种非二利液体的模型，其中$ρ$从真空中限制为$ρ$。一方面，我们通过引入一些新变量来考虑一个精心设计的重新制定结构，实际上，这些变量可以将时间演变的变性和粘度转移到某些特殊源术语的可能奇点。 On the other hand, it is observed that, for the spherically symmetric flow, the radial projection of the so-called effective velocity $\boldsymbol{v} =U+\nabla φ(ρ)$ ($U$ is the velocity of the fluid, and $φ(ρ)$ is a function of $ρ$ defined via the shear viscosity coefficient $μ(ρ)$: $φ'（ρ）=2μ（ρ）/ρ^2 $）验证一个阻尼的传输方程，该方程提供了获得其上限的可能性。然后与BD熵估计相结合，可以获得所需的溶液的先验估计。值得指出的是，此处建立的适当性理论的框架工作可以应用于浅水方程。
We consider the initial-boundary value problem (IBVP) for the isentropic compressible Navier-Stokes equations (\textbf{CNS}) in the domain exterior to a ball in $\mathbb R^d$ $(d=2\ \text{or} \ 3)$. When viscosity coefficients are given as a constant multiple of the mass density $ρ$, based on some analysis of the nonlinear structure of this system, we prove the global existence of the unique spherically symmetric classical solution for (large) initial data with spherical symmetry and far field vacuum in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained have the conserved total mass and finite total energy. $ρ$ keeps positive in the domain considered but decays to zero in the far field, which is consistent with the facts that the total mass is conserved, and \textbf{CNS} is a model of non-dilute fluids where $ρ$ is bounded away from the vacuum. To prove the existence, on the one hand, we consider a well-designed reformulated structure by introducing some new variables, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. On the other hand, it is observed that, for the spherically symmetric flow, the radial projection of the so-called effective velocity $\boldsymbol{v} =U+\nabla φ(ρ)$ ($U$ is the velocity of the fluid, and $φ(ρ)$ is a function of $ρ$ defined via the shear viscosity coefficient $μ(ρ)$: $φ'(ρ)=2μ(ρ)/ρ^2$), verifies a damped transport equation which provides the possibility to obtain its upper bound. Then combined with the BD entropy estimates, one can obtain the required uniform a priori estimates of the solution. It is worth pointing out that the frame work on the well-posedness theory established here can be applied to the shallow water equations.
