Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid.

*(English)*Zbl 1018.76012The 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.

Reviewer: Isabell Gallagher (Palaiseau)

##### 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 |