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Lack of collision between solid bodies in a 2D incompressible viscous flow. (English) Zbl 1221.35279
The motion of a solid body $$B$$ inside a cavity $$\Omega\subset \mathbb{R}^2$$ filled with an incompressible constant-density viscous fluid is under consideration. In absence of external forces, the action of the fluid causes the body to move. Usually, the incompressible Navier-Stokes equations (resp. the solid mechanics equations) in order to describe the behaviour of the fluid (resp. the body) are used. Here this system named full Navier-Stokes for short interactions between the body and the fluid holds in two ways: the incompressible Navier-Stokes equations are supplemented with Dirichlet boundary conditions over the interface fluid/body and the force and torque applied by the fluid on the body are taken into account in solid mechanics relations.
The main result in the study is: Assume $$\Omega$$ is a half-space in $$\mathbb{R}^2$$ and $$B$$ is a disk with radius 1. Furthermore, assume there is no external force. Then, any strong solution is global. In particular, for any strong solution, there is no collision between $$B$$ and the boundary of $$\Omega$$.

##### MSC:
 35Q30 Navier-Stokes equations 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 35R35 Free boundary problems for PDEs
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