Non-stationary Navier-Stokes Equations with Mixed Boundary Conditions

TL;DR

This paper proves the existence and uniqueness of solutions for the 2D and 3D Navier-Stokes equations with mixed boundary conditions under small initial data, extending previous results with weaker smoothness requirements.

Contribution

It introduces a new existence and uniqueness result for non-stationary Navier-Stokes equations with mixed boundary conditions, relaxing smoothness assumptions on initial data.

Findings

Unique solutions exist for small initial data.

Solutions depend continuously on perturbed data.

Weaker smoothness conditions improve applicability.

Abstract

In this paper we are concerned with the initial boundary value problem of the 2, 3-D Navier-Stokes equations with mixed boundary conditions including conditions for velocity, static pressure, stress, rotation and Navier slip condition together. Under a compatibility condition at the initial instance it is proved that for the small data there exists a unique solution on the given interval of time. Also, it is proved that if a solution is given, then there exists a unique solution for small perturbed data satisfying the compatibility condition. Our smoothness condition for initial functions in the compatibility condition is weaker than one in such a previous result.