On Well-posedness of a Nonstationary Stokes Hemivariational Inequality

TL;DR

This paper refines the well-posedness analysis of a nonstationary Stokes hemivariational inequality, weakening assumptions and exploring the equivalence of related inequalities for incompressible fluid flow with nonsmooth boundary conditions.

Contribution

It provides a more natural and less restrictive well-posedness analysis for the problem, including a refined approach for velocity and pressure fields and the study of multiple hemivariational inequalities.

Findings

Solution existence for velocity via semi-discrete approximations

Pressure recovered using inf-sup property

Weakened assumptions on source term and initial velocity

Abstract

This paper is devoted to the well-posedness analysis of a nonstationary Stokes hemivariational inequality for an incompressible fluid flow described by the Stokes equations subject to a nonsmooth boundary condition of friction type described by the Clarke subdifferential. In a recent paper [19], well-posedness of the nonstationary Stokes hemivariational inequality is studied for both the velocity and pressure fields. The solution existence is shown through a limiting procedure based on temporally semi-discrete approximations for both the velocity and pressure fields. In this paper, a refined well-posedness analysis is provided on the nonstationary Stokes hemivariational inequality under more natural assumptions on the problem data. The solution existence is first shown for the velocity field through a limiting procedure based on temporally semi-discrete approximations of a reduced problem and then the pressure field is recovered with the help of an inf-sup property. In this way, assumptions on the source term and the initial velocity needed in [19] are weakened, and a compatibility condition on initial values of the data is dropped. Moreover, several hemivariational inequalities are introduced for the mathematical model and their equivalence is explored.