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The Stokes problem and vector potential operator in three-dimensional exterior domains: An approach in weighted Sobolev spaces. (English) Zbl 0831.35125
This paper is concerned with some aspects of flows around bounded obstacles $$\Omega'$$ with Lipschitz boundary $$\Gamma$$. Considered is the three dimensional Stokes problem $- \nu \Delta \vec u + \nabla p = \vec f,\;- \text{div} \vec u = h$ in the exterior domain $$\Omega$$ together with $$\vec u = \vec g$$ on $$\Gamma$$. Firstly, the author presents a short review of earlier works in this field where mainly $$L^p$$ spaces have been introduced.
Here, the data are supposed to belong to certain weighted Sobolev spaces, and the solution or the vector potential is also sought in appropriate weighted Sobolev spaces. The results include theorems on equivalent norms, an inf-sup condition for the divergence, characterization of vector potentials of divergence free vector fields, and finally on existence, uniqueness and regularity of the solution to the Stokes problem.

##### MSC:
 35Q30 Navier-Stokes equations 35D10 Regularity of generalized solutions of PDE (MSC2000) 35D05 Existence of generalized solutions of PDE (MSC2000) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 76D07 Stokes and related (Oseen, etc.) flows