Communications in Mathematical Physics

, Volume 192, Issue 2, pp 463–491

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

DOI: 10.1007/s002200050305

Cite this article as:
Sammartino, M. & Caflisch, R. Comm Math Phys (1998) 192: 463. doi:10.1007/s002200050305

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Marco Sammartino
    • 1
  • Russel E. Caflisch
    • 2
  1. 1.Dipartimento di Matematica, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy.¶E-mail: marco@ipamat.math.unipa.itIT
  2. 2.Mathematics Department, UCLA, Los Angeles, CA 90096-1555, USA. E-mail: caflisch@math.ucla.eduUS