Global well-posedness of solutions for the equations modelling the motion of a rigid body in a bidimensional perfect fluid

TL;DR

This paper proves the global well-posedness of solutions for equations modeling a rigid body's motion in a 2D perfect fluid, extending previous results to arbitrary shapes and removing certain initial data constraints.

Contribution

It generalizes prior work by removing shape restrictions and initial vorticity constraints, providing a broader well-posedness result for the system.

Findings

Established a Beale-Kato-Majda type bound for the system.

Proved global existence and uniqueness of solutions for arbitrary-shaped rigid bodies.

Derived an explicit energy bound for the solutions.

Abstract

This paper considers a system modelling the evolution of a rigid body immersed in a bidimensional incompressible perfect fluid. In the special case of a disk-shaped rigid body, it was shown by C. Rosier and L. Rosier (2009) that the system admits a unique global solution when the initial fluid velocity $u_0$ belongs to $H^s$ ($s \ge 3$) and its vorticity $\operatorname{curl} u_0$ lies in $L^p$ with $1 \le p < 2$. By establishing a Beale-Kato-Majda type bound, we generalize the result by removing the constraint $\operatorname{curl} u_0 \in L^p$ and allowing the rigid body to be of arbitrary shape. Moreover, we obtain an explicit energy bound.