Communications in Mathematical Physics

, Volume 192, Issue 2, pp 433-461

First online:

Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.¶I. Existence for Euler and Prandtl Equations

  • Marco SammartinoAffiliated withDipartimento di Matematica, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy.¶E-mail: marco@ipamat.math.unipa.it
  • , Russel E. CaflischAffiliated withMathematics Department, UCLA, Los Angeles, CA 90096-1555, USA. E-mail: caflisch@math.ucla.edu

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Abstract:

This is the first of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the projection method is used in the primitive variables, to which the Cauchy-Kowalewski theorem is directly applicable. For the Prandtl equations, Cauchy-Kowalewski is applicable once the diffusion operator in the vertical direction is inverted.