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Local and Global Solvability of Free Boundary Problems for the Compressible Navier–Stokes Equations Near Equilibria

  • Irina Vladimirovna Denisova
  • Vsevolod Alexeevich Solonnikov
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Abstract

The chapter is concerned with free boundary and interface problems for equations governing viscous compressible flow. The main difficulty of such problems is due to the fact that the surface of the fluid is unknown. A proof of the classical solvability is outlined of the problem on the motion of a drop in vacuum in a finite time interval both in the case of the presence of surface tension on the free boundary and without it. The motion of two compressible fluids and fluids of different types, compressible and incompressible, separated by an unknown interface is also studied. For the latter problem, the global-in-time solvability in the Sobolev–Slobodetskiı̌ spaces \(W_2^{l,l/2}\) is proved in the case where surface tension is not taken into account and the data are small. The basic tools of analysis of free boundary problems are the passage to Lagrangian coordinates, the Fourier–Laplace transform, and the Plancherel theorem. An exponential energy inequality is also obtained; it is applied to show global existence and exponential decay of a solution in the Sobolev–Slobodetskiı̌ spaces. In addition, some results of potential theory are used in studying Hölder continuous solutions.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Irina Vladimirovna Denisova
    • 1
  • Vsevolod Alexeevich Solonnikov
    • 2
  1. 1.Laboratory for Mathematical Modelling of Wave PhenomenaInstitute for Problems in Mechanical Engineering of the Russian Academy of SciencesSt. PetersburgRussia
  2. 2.Laboratory of Mathematical PhysicsSt. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of SciencesSt. PetersburgRussia

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