On steady Stokes and Navier-Stokes problems with zero velocity at infinity in a three-dimensional exterior domain. (English) Zbl 0976.35051

Nazarov, S. A.; Pileckas, K.

J. Math. Kyoto Univ. 40, No. 3, 475-492 (2000).

This paper deals with the exterior Stokes and Navier-Stokes problems with zero velocity at infinity and in weighted function spaces. The authors construct in the case of small data the solution of the Navier-Stokes problem which has an appropriate asymptotic representation. The vanishing at infinity velocity field is prescribed on the boundary. A special attention is given to the derivation of the asymptotic formulae.
The Stokes problem is studied in weighted spaces with detached asymptotics and it is shown that the Stokes operator is of Fredholm type in these spaces.
The Banach contraction principle enables the authors to prove the solvability of the (nonlinear !) exterior Navier-Stokes problem in the case of small data. The “large” data case remains open.

Reviewer: Petar Popivanov (Sofia)

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

exterior Stokes and Navier-Stokes problems; weighted function spaces; small data; asymptotic representation; Stokes operator; solvability