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In this chapter, we consider the existence of the finite energy weak solution to the equations of a barotropic flow inside the time dependent domain. The proof is presented for Navier slip boundary conditions for the fluid velocity. The main difference with respect to Chap. 3 is that the penalization term is supported only on the boundary of the moving domain. This allows to conclude that not only the equations of motion but also the energy inequality is satisfied for the limit functions. The theorem stating the existence of a finite energy weak solution to the problem with Dirichlet boundary condition is stated here as well, and its proof is a straightforward modification of the one for the Navier slip boundary conditions.

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Barotropic Viscous Fluid with Slip Boundary Condition

  • 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

In this chapter, we consider the existence of the finite energy weak solution to the equations of a barotropic flow inside the time dependent domain. The proof is presented for Navier slip boundary conditions for the fluid velocity. The main difference with respect to Chap. 3 is that the penalization term is supported only on the boundary of the moving domain. This allows to conclude that not only the equations of motion but also the energy inequality is satisfied for the limit functions. The theorem stating the existence of a finite energy weak solution to the problem with Dirichlet boundary condition is stated here as well, and its proof is a straightforward modification of the one for the Navier slip boundary conditions.