具小粘性的可压缩Navier-Stokes方程的整体解
Global solutions of compressible Navier-Stokes equations with small viscosity
AI总结：
本文针对三维可压缩Navier-Stokes系统柯西问题，在特定初始扰动条件下建立经典解整体存在性，改进了Matsumura-Nishida的结果，相关方法可推广至Shizuta-Kawashima系统。
AI中文摘要：
本文研究三维空间中可压缩Navier-Stokes系统的柯西问题。假设粘性系数满足0<max{μ, ν=λ+2μ}<1，记ε=min{μ, ν=λ+2μ}，当初始密度扰动、速度的无旋部分扰动小于ε^(1/2+)（允许对数损失），且初始速度的散度部分扰动小于ε时，我们建立了经典解的整体存在性。这改进了Matsumura-Nishida的经典整体存在性结果，该结果要求所有初始数据均小于ε（<1）。我们期望该结果能代表物理应用中产生的一般Shizuta-Kawashima系统。将指标从1改进为1/2+，依赖于利用密度的隐藏Kawashima型耗散，并以√ε的尺度控制解的时空迹范数；这两个要素通过加权迹不等式和针对扰动声速与速度散度的Morawetz型不等式得到。
英文摘要：
In this paper, we study the Cauchy problem for the compressible Navier-Stokes system in $\mathbb{R}^3$. Suppose that the viscosity coefficients satisfy $0<\max\{μ, ν=λ+2μ\}<1$, and set $\varepsilon=\min\{μ, ν=λ+2μ\}$. We establish the global existence of classical solutions when the initial perturbations of the density and the curl-free part of the velocity are smaller than $\varepsilon^{\frac12+}$ (up to a logarithmic loss), while the divergence part of the initial velocity is smaller than $\varepsilon$. This improves the classical global existence result of Matsumura-Nishida \cite{MaN80}, which requires all the initial data to be smaller than $\varepsilon (<1)$. We expect that this result is representative of general Shizuta-Kawashima systems arising in physical applications. The improvement of the index from $1$ to $\frac12+$ relies on exploiting the hidden Kawashima-type dissipation for the density and controlling the spacetime trace norm of the solution at the scale $\sqrt{\varepsilon}$. These two ingredients are obtained through a weighted trace inequality and a Morawetz-type inequality for the perturbed sound speed and the divergence of the velocity.
