The steady Navier-Stokes and Stokes systems with mixed boundary conditions including one-sided leaks and pressure

TL;DR

This paper develops variational formulations and proves existence and uniqueness of solutions for steady Navier-Stokes and Stokes systems with complex mixed boundary conditions, including leaks and slip, without dependence on threshold parameters.

Contribution

It introduces new variational inequalities for these boundary conditions and establishes solution existence, uniqueness, and estimates independent of slip and leak thresholds.

Findings

Established variational inequalities for complex boundary conditions.

Proved existence and uniqueness of solutions.

Derived estimates independent of slip and leak thresholds.

Abstract

In this paper we are concerned with the steady Navier-Stokes and Stokes problems with mixed boundary conditions involving Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, rotation, stress and normal derivative of velocity together. Relying on the relations among strain, rotation, normal derivative of velocity and shape of boundary surface, we have variational formulations for the problems, which consist of five formulae with five unknowns. We get the variational inequalities equivalent to the formulated variational problems, which have one unknown. Then, we study the corresponding variational inequalities and relying the results for variational inequalities, we get existence, uniqueness and estimates of solutions to the Navier-Stokes and Stokes problems with the boundary conditions. Our estimates for solutions do not depend on the thresholds for slip and leaks.