论文标题
三维压缩的Navier-Stokes方程的消失粘度极限，具有退化粘度和远场真空
Vanishing Viscosity Limit of the Three-Dimensional Barotropic Compressible Navier-Stokes Equations with Degenerate Viscosities and Far-Field Vacuum
论文作者
论文摘要
我们关注的是，在$ \ Mathbb {r}^3 $中，Navier-Stokes方程的navier-Stokes方程限制到Euler方程的欧拉方程。当粘度系数遵守密度的较低幂（即，$ρ^δ$带有$ 0 <δ<1 $）时，我们确定了Navier-Stokes方程式的准单抗椭圆形耦合结构，以控制液体液体速度的行为。然后，采用这种结构来证明，有一个独特的常规解决方案，这些解决方案是对相应的考奇问题的独特解决方案，并使用任意大的初始数据和远场真空，其寿命在消失的粘度限制下均匀地为正。还获得了$ h^3（\ Mathbb {r}^3）$在粘度系数方面的一些统一估计，并获得了纳维尔 - 长方形方程的常规解决方案的强烈融合，并具有有限的质量和能量的eulere eulere eulere eulere sequ和euler eulere eulere in $ l^n $ l^^eftty in $ l^fift的定期解决方案。 h^{s} _ {\ rm loc}（\ mathbb {r}^3））$ for [2，3）$中的任何$ s \ for notey $ s \ for [2，3）$。结果，我们表明，对于粘性和无粘性流，任何全球常规解决方案的$ l^\ infty $ norm in Vacuum nord in Vacuum nords均可逐渐渐近，因为$ t $倾向于无限。我们在此开发的框架适用于通过一些次要修改的其他物理维度的相同问题。
We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for barotropic compressible fluids in $\mathbb{R}^3$. When the viscosity coefficients obey a lower power-law of the density (i.e., $ρ^δ$ with $0<δ<1$), we identify a quasi-symmetric hyperbolic--singular elliptic coupled structure of the Navier-Stokes equations to control the behavior of the velocity of the fluids near the vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field vacuum, whose life span is uniformly positive in the vanishing viscosity limit. Some uniform estimates on both the local sound speed and the velocity in $H^3(\mathbb{R}^3)$ with respect to the viscosity coefficients are also obtained, which lead to the strong convergence of the regular solutions of the Navier-Stokes equations with finite mass and energy to the corresponding regular solutions of the Euler equations in $L^{\infty}([0, T]; H^{s}_{\rm loc}(\mathbb{R}^3))$ for any $s\in [2, 3)$. As a consequence, we show that, for both viscous and inviscid flows, it is impossible that the $L^\infty$ norm of any global regular solution with vacuum decays to zero asymptotically, as $t$ tends to infinity. Our framework developed here is applicable to the same problem for the other physical dimensions via some minor modifications.
