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The aim of this chapter is to prove the existence of a weak solution to the compressible barotropic Navier-Stokes system with no-slip boundary condition for fluid velocity on domains with boundaries varying in time. We present the proof using the Brinkman-type penalization method which allows us to conclude that the limit functions satisfy the equations of motion. The method however does not allow us to conclude that the energy inequality is satisfied; therefore we only prove the existence of a weak solution here, not the existence of finite energy weak solution. The purpose of this chapter is mainly to illustrate the limit procedure inside the fluid domain, which will be similar in the proofs in the following chapters.

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Barotropic Viscous Fluid with the Dirichlet Boundary Conditions

  • 章节

  • 发表日期：

    2025年

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  • Ondřej Kreml,
  • Václav Mácha,
  • Šárka Nečasová,
  • Tomasz Piasecki,
  • Aneta Wróblewska-Kamińska

摘要

The aim of this chapter is to prove the existence of a weak solution to the compressible barotropic Navier-Stokes system with no-slip boundary condition for fluid velocity on domains with boundaries varying in time. We present the proof using the Brinkman-type penalization method which allows us to conclude that the limit functions satisfy the equations of motion. The method however does not allow us to conclude that the energy inequality is satisfied; therefore we only prove the existence of a weak solution here, not the existence of finite energy weak solution. The purpose of this chapter is mainly to illustrate the limit procedure inside the fluid domain, which will be similar in the proofs in the following chapters.