Existence of weak solutions for the motion of rigid bodies in a viscous fluid. (English) Zbl 0943.35063

Desjardins, B.; Esteban, M. J.

Arch. Ration. Mech. Anal. 146, No. 1, 59-71 (1999).

The local existence of strong solutions for the interaction of a rigid disk with a two-dimensional viscous incompressible flow is studied in [C. Grandmont and Y. Maday, C. R. Acad. Sci. Paris, Sér. I, Math. 326, No. 4, 525-530 (1998; Zbl 0924.76022)]. In the present paper, a global weak formulation of the above problem is introduced, for a finite number of rigid bodies in interaction with an incompressible viscous fluid, in a bounded domain with dimension 2 or 3. The existence of solutions for a particular form of the initial velocities is proved. In the absence of collisions, a global in time solution exists in the plane; in dimension 3 it is necessary to have small enough data. The main part of the proof involves the Di Perna-Lions theorem on compactness of sequences of solutions to linear transport equations.

Reviewer: G.Paşa (Bucureşti)

MSC:

35Q30	Navier-Stokes equations
35D05	Existence of generalized solutions of PDE (MSC2000)
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids

Keywords:

small data; weak formulation; bounded domain; compactness

Citations:

Zbl 0924.76022