Existence of the solution to stationary Navier–Stokes equations with nonlinear slip boundary conditions

Abstract

The stationary Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper. Because the boundary conditions include the subdifferential property on the part boundary, the variational formulation of this problem is the variational inequality problem of the second kind with Navier-Stokes operator. The main purpose of the paper is to study the existence of the weak solution and the strong solution to this variational inequality problem in terms of the Yosida's regularity method.

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

“…This friction type of boundary condition such as leak and slip boundary involving subdifferential property has first been analyzed by Fujita et al , where he researched some hydrodynamics problems. These theoretical problems were also studied by many scholars, such as Fujita et al , Saito et al , Li et al , and references cited therein. Other numerical theory results about steady and time‐dependent Navier‐Stokes problems with friction boundary conditions can be found in .…”

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

“…Subsequently, Saito studied regularity of the weak solution for the Stokes equations by Yosidas regularized method and difference quotients. Meanwhile, the existence and uniqueness of the weak solution to the steady and unsteady Navier‐Stokes problems under friction boundary conditions were obtained. For numerical methods for the steady Navier‐Stokes equations, Li and An studied penalty finite element method for this type of boundary conditions .…”

A penalty‐FEM for navier‐stokes type variational inequality with nonlinear damping term

“…Some theoretical problems were also investigated by many scholars (see Refs. [3][4][5][6][7]). Other numerical theory results about Navier-Stokes problems with friction type boundary conditions can be found in Refs.…”