On the K-th Power of Stokes Operator

Stokes operators, are well known partial differential operators of elliptic type, which are often used in Applied Mathematics. Stokes equation describes the irrotational, axisymmetric creeping flow and Stokes bi-stream equation denotes the rotational one, where Necessary and sufficient conditions for the separability and the R-separability of the equation have been proved recently. Moreover, the 0-eigenspace and the generalized 0-eigenspace of the operator have been derived in several coordinate systems. Specifically, the spherical coordinate system is employed in many problems taking into account that in many engineering applications, the solutions in spherical geometry seem to be adequate for solving a problem. In the present manuscript, it is shown that equation admits a solution of the form where are solutions of Stokes equation and r is the radial spherical variable. Additionally, we obtain the kernel of the k-th power of the Stokes operator, in the spherical geometry for every