The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Incompressible Euler as a limit of complex fluid models with Navier boundary conditions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)In this article we study the limit alpha -> 0 of solutions of the alpha-Euler equations and the limit alpha, v -> 0 of solutions of the second grade fluid equations in a bounded domain, both in two and in three space dimensions. We prove that solutions of the complex fluid models converge to solutions of the incompressible Euler equations in a bounded domain with Navier boundary conditions, under the hypothesis that there exists a uniform time of existence for the approximations, independent of a and v. This additional hypothesis is not necessary in 2D, where global existence is known, and for axisymmetric flows without swirl, for which we prove global existence. Our conclusion is strong convergence in L(2) to a solution of the incompressible Euler equations, assuming smooth initial data. (C) 2011 Elsevier Inc. All rights reserved.2521624640Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [303.301/2007-4, 302.214/2004-6]FAPESP [2007/51490-7]FAPESP [22076