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2D Navier–Stokes Equations in a Time Dependent Domain with Neumann Type Boundary Conditions

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In this paper the two-dimensional Navier–Stokes system for incompressible fluid coupled with a parabolic equation through the Neumann type boundary condition for the second component of the velocity is considered. Navier–Stokes equations are defined on a given time dependent domain. We prove the existence of a weak solution for this system. In addition, we prove the continuous dependence of solutions on the data for a regularized version of this system. For a special case of this regularized system also a problem with an unknown interface is solved.

The problem under consideration is an approximation of the fluid-structure interaction problem proposed by A. Quarteroni in [19]. We conjecture that our approach is useful also for the numerical treatment of the problem and at the end we shortly present our numerical experiments.

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Communicated by H. Beirão da Veiga

Anna Zaušková: Supported by the Grants of the Comenius University 66/2004 and 321/2005.

Both authors were supported by the VEGA Grant Agency of the Slovak Republic No. 1/0260/03.

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Filo, J., Zaušková, A. 2D Navier–Stokes Equations in a Time Dependent Domain with Neumann Type Boundary Conditions. J. Math. Fluid Mech. 12, 1–46 (2010).

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Mathematics Subject Classification (2000).