Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

✍ Scribed by Helmut Abels

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 256 KB
• Volume: 279
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310365

✦ Synopsis

Abstract

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω~0~ = ℝ^n –1^ × (–1, 1), n ≥ 2, in L^q^ ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω^+^~0~ = ℝ^n –1^ × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–^~0~ = ℝ^n –1^ × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q^ ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ^2^‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)