An estimate for solutions of the Stokes equations in exteriour domains

P. Maremonti   ; V. A. Solonnikov

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 9, Tome 180 (1990), pp. 105-120

Résumé

A boundary value problem for the Stokes equations in an exterior domain $\Omega\subset\mathbb{R}^n$ with the Dirichlet condition on the boundary and a homogeneous condition at the infinity for the velocity vector field is considered. It is shown that the $L(p)-$norm of the $2^{\textrm{nd}}$ derivatives of this vector field is estimated by the same norm of the exterior forces vector field. This estimate holds for $p<\frac n2$; for $p\geqslant\frac n2$ it is valid only for a problem with modified conditions at the infinity.