JournalszaaVol. 20, No. 1pp. 155–182

On the Fredholm Property of the Stokes Operator in a Layer-Like Domain

  • Sergei A. Nazarov

    Institute for Problems in Mechanical Engineering RAS, St. Petersburg, Russian Federation
  • K. Pileckas

    Vilnius University, Lithuania
On the Fredholm Property of the Stokes Operator in a Layer-Like Domain cover

Abstract

The Stokes problem is studied in the domain ΩR3\Omega \subset \mathbb R^3 coinciding with the layer Π=x=(y,z):y=(y1,y2)(0,1)\Pi = {x = (y,z) : y = (y_1, y_2) \in (0,1)} outside some ball. It is shown that the operator of such problem is of Fredholm type; this operator is defined on a certain weighted function space Dβl(Ω­\mathcal D^l_{\beta} (\Omega­) with norm determined by a stepwise anisotropic distribution of weight factors (the direction of zz is distinguished). The smoothness exponent ll is allowed to be a positive integer, and the weight exponent β\beta is an arbitrary real number except for the integer set Z\mathbb Z where the Fredholm property is lost. Dimensions of the kernel and cokernel of the operator are calculated in dependence of β\beta. It turns out that, at any admissible β\beta, the operator index does not vanish. Based on the generalized Green formula, asymptotic conditions at infinity are imposed to provide the problem with index zero.

Cite this article

Sergei A. Nazarov, K. Pileckas, On the Fredholm Property of the Stokes Operator in a Layer-Like Domain. Z. Anal. Anwend. 20 (2001), no. 1, pp. 155–182

DOI 10.4171/ZAA/1008