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A generalized Stokes system with a non-smooth slip boundary condition.

Jing Zhao1, Stanislaw Migórski1,2, Sylwia Dudek3

  • 1College of Sciences, Beibu Gulf University, Qinzhou, Guangxi 535000, People's Republic of China.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|September 26, 2022
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Summary

This study establishes well-posedness for quasi variational-hemivariational inequalities with convex potentials and non-monotone constraints. Results are applied to generalized Stokes models for incompressible fluids.

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Bingham-type fluidStokes equationgeneralized subgradientslip conditionvariational–hemivariational inequality

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Area of Science:

  • Mathematical analysis
  • Applied mathematics
  • Fluid dynamics

Background:

  • Quasi variational-hemivariational inequalities are crucial in modeling complex physical phenomena.
  • Understanding their well-posedness is essential for reliable simulations and predictions.
  • Existing models often struggle with non-monotone conditions and implicit constraints.

Purpose of the Study:

  • To investigate a class of quasi variational-hemivariational inequalities in reflexive Banach spaces.
  • To establish conditions for the well-posedness of these inequalities.
  • To demonstrate the practical relevance through an application to fluid dynamics.

Main Methods:

  • Utilizing concepts from convex analysis and non-smooth analysis.
  • Employing fixed-point theorems and variational methods.
  • Analyzing the properties of convex potentials and locally Lipschitz superpotentials.

Main Results:

  • Existence and uniqueness of solutions are proven.
  • The continuous dependence of solutions on the given data is established.
  • The compactness of the solution set in the strong topology is demonstrated.

Conclusions:

  • The theoretical results provide a robust framework for analyzing complex inequality problems.
  • The study successfully applies these findings to the generalized Stokes model.
  • This work contributes to the understanding of non-smooth variational problems in applied mathematics.