Low regularity solutions for the two-dimensional “rigid body + incompressible Euler” system. (English) Zbl 1340.76018

Glass, Olivier; Sueur, Franck

Differ. Integral Equ. 27, No. 7-8, 625-642 (2014).

In this paper, the authors consider the motion of a solid body in an incompressible perfect fluid filling the plane in a low regularity setting. They assume that \(S_0\) is a closed, bounded, connected subset of the plane with smooth boundary. Moreover, they consider that the body initially occupies the domain \(S_0\) and rigidly moves so that at time \(t\) it occupies an isometric domain denoted by \(S(t)\). They denote by \(F(t)\) the domain occupied by the fluid at time \(t\) starting from the initial domain \(F(t)=\mathbb R^2 \setminus S_0\). The motion of the fluid is described by incompressible Euler equations and the motion of a rigid body is done by Newton’s balance law for linear and angular momenta.
A quantity which plays a central role is the vorticity field \(w(t,x)=\text{curl} \,u(t,x)\).
The main goal is to prove the global existence of some solutions of this system in the case \(u_0 \in L^2\). They underline that \(u_0\) is not necessarily of finite energy. A general strategy for obtaining a weak solution is to regularize the initial data so that one gets a sequence of initial data which generate some classical solutions and then to pass to the limit with respect to the regularization parameter in the weak formulation of the equations.

Reviewer: Šárka Nečasová (Praha)

MSC:

76B99	Incompressible inviscid fluids
74F10	Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)

Keywords:

incompressible Euler equations; global existence; weak solution