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

Xiaoguang You

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.

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

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 belongs to () and its vorticity lies in with . By establishing a Beale-Kato-Majda type bound, we generalize the result by removing the constraint and allowing the rigid body to be of arbitrary shape. Moreover, we obtain an explicit energy bound.
Paper Structure (11 sections, 16 theorems, 130 equations)

This paper contains 11 sections, 16 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Poisson equation in an exterior domain
  4. Part 1. Estimating $\nabla \psi$
  5. Part 2. Estimating $D^2 \psi$
  6. Step 1. Estimates on $\mathbb{R}^2$
  7. Step 2. Estimates on $\mathcal{C}_{R_3}$
  8. Step 3. Merge the estimates
  9. A priori estimates
  10. Proof of the main theorem
  11. Acknowledgments

Key Result

Theorem 1.1

Let $s \geqslant 3$ be an integer. Suppose that $u_0 \in H^s(\mathcal{F})$ satisfies Then the system Equ:euler-1--Equ:euler-4 admits a unique solution $(u, p, h, r) \in \mathscr X_{s;\infty}$. Moreover, there exist constants $K_i$ ($i=1,2,3$), independent of $s$, such that for all $t > 0$, Here $\lambda(\tau)$ does not depend on $s$ and is given by where $\ln^+ x = \max(0,\ln x)$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Proposition 3.1: Existence
  • proof
  • Proposition 3.2: Regularity
  • ...and 17 more