Zero viscosity limit for analytic solutions of the navier-stokes equation on a half-space. II. Construction of the Navier-stokes solution

Marco Sammartino, Russel E. Caflisch

Research output: Contribution to journalArticle

Abstract

This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.

Original languageEnglish (US)
Pages (from-to)463-491
Number of pages29
JournalCommunications In Mathematical Physics
Volume192
Issue number2
DOIs
StatePublished - 1998

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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