Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain

TL;DR

This paper proves the existence and uniqueness of strong solutions to 2D Navier-Stokes equations with Navier-slip boundary conditions in a strip domain, even with indefinite slip coefficients, and establishes useful inequalities.

Contribution

It introduces new results on well-posedness for Navier-Stokes with slip boundary conditions and develops Gagliardo-Nirenberg inequalities for related Sobolev spaces.

Findings

Existence and uniqueness of strong solutions in a 2D strip domain.

Applicability of Gagliardo-Nirenberg inequalities to Navier-Stokes problems.

Handling of indefinite slip coefficients in boundary conditions.

Abstract

This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.