Lp-THEORY FOR VECTOR POTENTIALS AND SOBOLEV'S INEQUALITIES FOR VECTOR FIELDS: APPLICATION TO THE STOKES EQUATIONS WITH PRESSURE BOUNDARY CONDITIONS

Abstract

In a three-dimensional bounded possibly multiply connected domain, we give gradient and higher-order estimates of vector fields via div and curl in Lp-theory. Then, we prove the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, we consider the stationary Stokes equations with non-standard boundary conditions of the form u × n = g × n and π = π0on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are based on obtaining Inf–Sup conditions that play a fundamental role. We give a variant of the Stokes system with these boundary conditions, in the case where the compatibility condition is not verified. Finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as u⋅n and u × n on Γ.