Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid

Abstract

Abstract. In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying R 2 . We prove the global in time existence and the uniqueness of the classical solution for this fluid-structure problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid-structure problem obtained by incorporating some dissipation.Mathematics Subject Classification. 35Q35, 76B03, 76B99.

A simple 1D model of inviscid fluid–solid interaction

“…Several works have dealt with the study of the system composed by rigid bodies and a viscous incompressible fluid; see, for instance, [6,7,13,[17][18][19][20]26,28,30,32,33]. Some authors have also considered structures of different type (see, for instance, [2,4,9]) or fluids of different type (see, for instance, [8,12,24]). However, up to now and up to our knowledge no results concerning the interaction between shocks and solids have been published.…”

Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

“…For potential flows the first studies of the problem (8)- (15) dates back to D'Alembert, Kelvin and Kirchoff. In the general case, the existence and uniqueness of classical solutions to the problem (8)- (15) is now well-understood thanks to the works of Ortega, Rosier and Takahashi [14]- [15], Rosier and Rosier [16] in the case of a body in R 3 and Houot, San Martin and Tucsnak [10] in the case (considered here) of a bounded domain, in Sobolev spaces H m , m 3.…”

“…In the last years, several papers have been devoted to the study of the dynamics of a rigid body immersed into a fluid governed by the Navier-Stokes equations. We refer to the introduction of [15] for a survey of these results.…”

Smooth solutions for the motion of a ball in an incompressible perfect fluid

“…If in the last decade a large number of papers have been devoted to the wellposedness of fluid-structure interaction problems involving a viscous fluid (that is, governed by Navier-Stokes equations), the motion of a rigid body in a (not potential) Eulerian flow has been investigated only in a few papers. In [11], the existence and uniqueness of a (global) classical solution of (1.1)-(1.8) was established when N = 2. A result in the same vein was obtained in [12] for a body of arbitrary form, again for N = 2.…”

“…A result in the same vein was obtained in [12] for a body of arbitrary form, again for N = 2. The aim of this paper is to extend the results of [11] to a space of arbitrary dimension N (N ∈ {2, 3} in practice), and to any order of smoothness. We shall for instance establish the existence of C ∞ smooth (global) solutions when N = 2.…”