In this chapter, we state the weak-strong uniqueness principle for the compressible barotropic Navier-Stokes system on a moving domain. This principle tells us that a weak solution coincides with a regular solution corresponding to the same data when the latter exists. To this end we first deduce the relative energy inequality for both the Dirichlet boundary condition and the Navier slip boundary condition. Then, we use the relative energy inequality to deduce the weak-strong uniqueness principle. An emphasis is laid upon the proof for the slip boundary conditions. Eventually, we comment on the differences arising when the Dirichlet boundary conditions are considered.

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Weak-Strong Uniqueness

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

摘要

In this chapter, we state the weak-strong uniqueness principle for the compressible barotropic Navier-Stokes system on a moving domain. This principle tells us that a weak solution coincides with a regular solution corresponding to the same data when the latter exists. To this end we first deduce the relative energy inequality for both the Dirichlet boundary condition and the Navier slip boundary condition. Then, we use the relative energy inequality to deduce the weak-strong uniqueness principle. An emphasis is laid upon the proof for the slip boundary conditions. Eventually, we comment on the differences arising when the Dirichlet boundary conditions are considered.