The boundary value problems of the Stokes resolvent equations in n dimensions

Abstract

We construct a solution for the boundary value problems of the Stokes resolvent system in bounded and exterior domains of R n (n ≥ 2) with prescribed Dirichlet-and Neumann boundary data. The construction is based on the explicit form of the corresponding fundamental and double layer tensor and uses hydrodynamical potential theory and the method of integral equations.

Existence and uniqueness result for the problem of viscous flow in a granular material with a void

“…Kohr [12] obtained a boundary integral method in order to study the oscillatory Stokes flow due to the oscillations of two solid spheres in an incompressible Newtonian fluid. On the other hand, Varnhorn [39], [40], [41] developed a complete potential theory for the Stokes resolvent system whose unknowns are defined only on domains in R n with connected boundaries. The method developed by Kohr [12] leads to the idea of the extension of the potential theory for the Stokes resolvent system from the case of domains in R n with connected boundaries to the general case of domains in R n (n ≥ 2) with non-connected boundaries (see [13], [14]).…”

The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in ℝn

“…Concluding the above statement we obtain the following result (see [34] in the case of a domain with connected boundary):…”

“…The following results refer to the continuity behaviour of the hydrodynamic potentials across the boundary Γ of the domain D (see [18, p. 201], [32, p. 66], [33] and [34]):…”

“…We have proved the following result (see [32, p. 76], [34] and [18, p. 214], in the case of bounded domains with connected boundaries):…”

“…Then we obtain the following result (see [32, p. 71], [33], and [34], in the case of an exterior domain with connected boundary):…”