Rate of Convergence in Inviscid Limit for 2D Navier-Stokes Equations with Navier Fricition Condition for Nonsmooth Initial Data

Abstract

AbstractWe are interested in the rate of convergence of solutions of 2D Navier-Stokes equations in a smooth bounded domain as the viscosity tends to zero under Navier friction condition. If the initial velocity is smooth enough(
“a—° Ii
Ž i
ŽðI), it is known that the rate of convergence is linearly propotional to the viscosity. Here, we consider the rate of convergence for nonsmoothvelocity fields when the gradient of the corresponding solution of the Euler equations belongs to certain Orlicz spaces. As acorollary, if the initial vorticity is bounded and small enough, we obtain a sublinear rate of convergence. Key words : Navier-Stokes, Inviscid Limit 1. Introduction The incompressible Navier-Ntokes equations are equations of motion of viscous fluid, while the incompressible Euler equations are that of nonviscous (ideal) fluid. The Cauchy problems for these equations are known to be well-defined if the initial velocity is smooth enough [1,2] . As a subsequent fundamental question, the relation between these two equations has been sought in the literature. That is, whether the solution of the Navier-Stokes equations converges to that of the Euler equations in a nice sense as the viscosity vanishes. This question of vanishing viscosity limit is relatively easy and answered affirmatively if there is no spatial boundary