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On Well-posedness of a Nonstationary Stokes Hemivariational Inequality

Weimin Han, Shengda Zeng

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.

On Well-posedness of a Nonstationary Stokes Hemivariational Inequality

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.

Paper Structure

This paper contains 8 sections, 15 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical model
  4. Reduced formulations for the velocity field
  5. Existence and uniqueness for the velocity field
  6. Existence and uniqueness of the pressure field
  7. Lipschitz continuous dependence on data
  8. Concluding remark

Key Result

Theorem 2.1

Let $1\leq p,q<\infty$. Let $V_1\subset V_2\subset V_3$ be real Banach spaces such that $V_1$ is reflexive, the embedding $V_1\subset V_2$ is compact and the embedding $V_2\subset V_3$ is continuous. Then a bounded subset of $M^{p,q}(I;V_1,V_3)$ is relatively compact in $L^p(I;V_2)$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Proposition 3.4
  • Proposition 3.7
  • Proposition 3.8
  • Proposition 3.11
  • Theorem 5.2
  • Remark 5.3
  • Lemma 5.4
  • ...and 6 more