Vanishing viscosity limit of Navier–Stokes Equations in Gevrey class

Abstract

In this paper, we consider the inviscid limit for the periodic solutions to Navier–Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier–Stokes equation is independent of viscosity, and that the solutions of the Navier–Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover, the convergence rate in Gevrey class is presented. Copyright © 2017 John Wiley & Sons, Ltd.

Uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in half-space

“…While, in the presence of physical boundaries, this problem becomes very challenging due to the appearance of boundary layers. For the no-slip boundary condition, the vanishing viscosity limit of the incompressible Navier-Stokes equations is wildly open except when the initial vorticity is located away from the boundary (see [23] on R 2 + ) or the initial data is analytic [20,[27][28][29][30][32][33][34], in Gevrey class [9], or have enough regularity [12]. For the Navier-slip boundary condition, the boundary layer is much weaker and more results are available, for example, see [3-6, 13, 17, 25, 31, 35, 38].…”

Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class

“…4279 4280 FUCAI LI AND ZHIPENG ZHANG phenomena governed by the viscous MHD equations. Thus, it is very interesting to consider the vanishing viscosity-resistivity limit of the viscous incompressible MHD equations.When we take H ν,µ = 0 in the system (1), it is reduced into the classical incompressible Navier-Stokes equations and many results are available on the inviscid limit to it in R n or torus, for instance, see [7,8,9,22,34] and the references cited therein. Specially, Cheng, Li, and Xu [7] studied the vanishing viscosity limit of the incompressible Navier-Stokes equations in Gevrey class in a torus and got the convergence rate.…”

“…In this paper, motivated by [7] on the incompressible Navier-Stokes equations, we study the zero viscosity-resistivity limit of the viscous incompressible MHD equations in the torus T 3 in Gevrey class. Due to the strong coupling between u ν,µ and H ν,µ , we need to estimate some new nonlinear terms.…”

“…Due to the strong coupling between u ν,µ and H ν,µ , we need to estimate some new nonlinear terms. In order to overcome the difficulty in handling the nonlinear terms, we shall establish a more general result (see Lemma 2.2 below) than that in Lemma 2.3 of [7]. Besides, we also need to construct more complicated energy estimates to obtain the desired convergence results.…”