The incompressible Navier-Stokes system with time-dependent Robin-type boundary conditions

TL;DR

This paper proves the existence of regular solutions to the 3D incompressible Navier-Stokes equations with time-dependent Robin boundary conditions in certain bounded domains, given small initial data.

Contribution

It introduces and analyzes a new boundary condition for the Navier-Stokes system, establishing solution existence and regularity under these conditions.

Findings

Existence of solutions with regularity under small initial data.

Applicability to domains with specific boundary conditions.

Extension of Navier-Stokes theory to time-dependent Robin boundaries.

Abstract

We show that the incompressible 3D Navier-Stokes system in a C 1;1 bounded domain or a bounded convex domain with a non penetration condition u = 0 at the boundary @ together with a time-dependent Robin boundary condition of the type curl u = (t)u on @ admits a solution with enough regularity provided the initial condition is small enough in an appropriate functional space.