Semi-group theory for the Stokes operator with Navier-type boundary conditions on $L^{p}$-spaces

TL;DR

This paper develops a semi-group framework for the Stokes operator with Navier boundary conditions on $L^{p}$ spaces, establishing analyticity, regularity, and maximal regularity results for solutions in complex geometries.

Contribution

It introduces a semi-group approach for the Stokes problem with Navier boundary conditions on non-simply connected domains, including fractional power analysis and maximal regularity results.

Findings

Proved analyticity of semigroups generated by the Stokes operator.

Established maximal regularity for the homogeneous Stokes problem.

Deduced $L^{p}$-$L^{q}$ regularity for the inhomogeneous problem.

Abstract

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $\Omega$, not necessarily simply connected. Since under these conditions the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypothesis on the initial data $\boldsymbol{u}_0$ and external force $\boldsymbol{f}$. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers we deduce maximal $L^{p}-L^{q}$ regularity for the inhomogeneous Stokes problem.