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Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. (English) Zbl 1018.76012
The authors study the motion of several inhomogeneous rigid bodies immersed in an incompressible inhomogeneous viscous fluid, all contained in a fixed open bounded subset of $$\mathbb{R}^2$$. The fluid is governed by Navier-Stokes equations, whereas the rigid bodies move following Newton’s laws for linear and angular momentum. The novelty compared with previous existence results on the subject is that collisions are allowed – however, the space dimension is restricted to two. The global existence of weak solutions satisfying an energy estimate is proved. Collisions can occur (between different solids or between a solid and the boundary) but only with vanishing relative velocity and acceleration. In order to prove the theorem, a penalized problem is introduced and solved, compactness being obtained by Di Perna-Lions theory. A special attention is paid to the possibility of collisions, namely in the proof of the compactness of velocity field.

##### MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 70E99 Dynamics of a rigid body and of multibody systems 35Q30 Navier-Stokes equations
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