论文标题
具有大数据和远处真空吸尘器的1-D退化可压缩的Navier-Stokes方程的全局常规解决方案
Global regular solutions for 1-D degenerate compressible Navier-Stokes equations with large data and far field vacuum
论文作者
论文摘要
在本文中，考虑了一维（1-D）等凝结的Navier-Stokes方程（\ textbf {CNS}）的Cauchy问题。当粘度$μ（ρ）$取决于均方根电力定律中的密度$ρ$（$ρ^δ$带有$ 0 <δ\ leq 1 $），基于对这种退化系统的内在奇异结构的详细分析，我们将具有一些定期良好的总体量的全球范围供应量的全球范围，并在整体上进行了良好的总体稳定量，并构成了稳定的势头，并有效地构成了稳定的势头，并且稳定的势头势力，并构成了稳定的势头，并量身定做。空间。此外，我们获得的解决方案满足了$ρ$在\ mathbb {r} $中的所有点$ x \保持阳性，但在远处却衰减为零，这与整个空间的总质量保存的事实是一致的，并且\ textbf {cns}是$ plature flature flat的模型。证明的关键是通过引入一些新的变量和初始兼容性条件来引入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脱落和粘度转移到某些特殊源术语的可能奇点。 Then, combined with the BD entropy estimates and transport properties of the so-called effective velocity $v=u+φ(ρ)_x$ ($u$ is the velocity of the fluid, and $φ(ρ)$ is a function of $ρ$ defined by $φ'(ρ)=μ(ρ)/ρ^2$), one can obtain the required uniform a priori estimates of corresponding solutions.
In this paper, the Cauchy problem for the one-dimensional (1-D) isentropic compressible Navier-Stokes equations (\textbf{CNS}) is considered. When the viscosity $μ(ρ)$ depends on the density $ρ$ in a sublinear power law ($ ρ^δ$ with $0<δ\leq 1$), based on an elaborate analysis of the intrinsic singular structure of this degenerate system, we prove the global-in-time well-posedness of regular solutions with conserved total mass, momentum, and finite total energy in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained satisfy that $ρ$ keeps positive for all point $x\in \mathbb{R}$ but decays to zero in the far field, which is consistent with the facts that the total mass of the whole space is conserved, and \textbf{CNS} is a model of non-dilute fluids where $ρ$ is bounded below away from zero. The key to the proof is the introduction of a well-designed reformulated structure by introducing some new variables and initial compatibility conditions, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. Then, combined with the BD entropy estimates and transport properties of the so-called effective velocity $v=u+φ(ρ)_x$ ($u$ is the velocity of the fluid, and $φ(ρ)$ is a function of $ρ$ defined by $φ'(ρ)=μ(ρ)/ρ^2$), one can obtain the required uniform a priori estimates of corresponding solutions.
