A New Approach to the Existence of Weak Solutions of the Steady Navier–Stokes System with Inhomogeneous Boundary Data in Domains with Noncompact Boundaries
- 127 Downloads
We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary ∂Ω consists of M unbounded components Γ1, . . . , Γ M and N − M bounded components ΓM+1, . . . , Γ N . We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L 3-extension to Ω with the gradient in L 2(Ω)3×3. We assume that the fluxes of α through the bounded components ΓM+1, . . . , Γ N of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ1, . . . , Γ M .
KeywordsWeak Solution Sobolev Inequality Unbounded Domain Lipschitzian Domain Stokes Problem
Unable to display preview. Download preview PDF.
- 4.Feistauer M.: Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics 67. Longman Scientific & Technical, Harlow, 1993Google Scholar
- 5.Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations vol. I, Linearized Steady Problems. Springer-Verlag, New York (1994)Google Scholar
- 7.Galdi, G.P.: Further properties of steady-state solutions to the Navier–Stokes problem past a three-dimensional obstacle. J. Math. Phys. 48, 065207, 43 pp (2007)Google Scholar
- 8.Kapitanskij C.J., Pileckas K.: On spaces of solenoidal vector fields and boundary value problems for the Navier–Stokes equations in domains with noncompact boundaries. Proc. Meth. Inst. Steklov 159(2), 3–34 (1994)Google Scholar
- 10.Kufner, A., John, O., Fučík, S.: Function Spaces. Academia, Prague, 1977Google Scholar
- 17.Zeidler E.: Nonlinear Functional Analysis and its Aplications I. Springer-Verlag, Berlin (1988)Google Scholar