New a priori estimates for the Stokes equations in exterior domains. (English) Zbl 0732.35068
The authors investigate the first boundary value problem for the stationary Stokes system in an exterior domain. Let \(\Omega\) be an exterior domain in \({\mathbb{R}}^ n\) (n\(\geq 3)\) with the smooth boundary \(\Gamma\). The following problem is considered in Sobolev spaces: find the vector \(u(x)=(u_ 1,...,u_ n)\) and the scalar function p(x) satisfying the Stokes equations
\[
-\Delta u+\nabla p=f,\quad div u=g\text{ in } \Omega
\]
and the boundary condition \(u|_{\Gamma}=\phi\). The existence and uniqueness of a weak solution \((\nabla u\in L_ q\), \(p\in L_ q)\) are proved. A priori estimates of this solution are established. The existence and uniqueness of a strong solution \((D^ 2u\in H^{m,q}\), \(\nabla p\in H^{m,q})\) are proved, too. Precise estimates of a strong solution are obtained. Proofs are based on methods of functional analysis.
Reviewer: I.Sh.Mogilevskij (Tver)
MSC:
35Q30 | Navier-Stokes equations |
35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |
35D05 | Existence of generalized solutions of PDE (MSC2000) |
35B65 | Smoothness and regularity of solutions to PDEs |