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Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible Fluid Mechanics

  • Vsevolod Alexeevich Solonnikov
  • Irina Vladimirovna Denisova
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Abstract

The chapter deals with unsteady motion of incompressible fluids with free boundaries and interfaces where the surface tension may be taken into account. Such flows are governed by free boundary problems for the Navier–Stokes system. For the problems on the motion of an isolated liquid mass in vacuum or in another fluid, local in time existence theorems are established in the Hölder and Sobolev–Slobodetskiı̌ classes of functions. The proofs are based on the estimates of solutions of the corresponding linear problems in fixed domains obtained by the Schauder localization and on constructing explicit solutions of the model problems in the dual Fourier–Laplace space.

The global solvability of some problems with small data is established by energy method. The solutions are proved to tend exponentially to those corresponding to equilibrium states.

Evolutionary problem on the stability of a rigid rotation of a viscous capillary drop with a given angular momentum is considered. The solution of the problem is proved to be exponentially stable in time if energy functional is positive. Instability theorem is also stated.

Some stationary free boundary problems are discussed. Among them there are problems on the periodic motion of a fluid above a rigid bottom and the motion of a fluid partially filling a container, the problem of filling a plane capillary with the contact angle equal to π. Existence theorems for these problems are stated.

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Authors and Affiliations

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

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