Existence of solutions for the equations modeling the motion of rigid bodies in an ideal fluid

In this paper, we study the motion of rigid bodies in a perfect incompressible fluid. The rigid-fluid system fils a bounded domain in R3. Adapting the strategy from Bourguignon and Brezis [1], we use the stream lines of the fluid and we eliminate the pressure by solving a Neumann problem. In this way, the system is reduced to an ordinary differential equation on a closed infinite dimensional manifold. Using this formulation, we prove the local in time existence and uniqueness of strong solutions. Notation. Throughout this paper Ω denotes an open bounded and connected subset of R and S0 is a closed set with nonempty interior and with smooth boundary such that S0 ⊂ Ω. We denote as usual by SO3(R) the special orthogonal group on R. We will often use functions defined from a time interval to R or to SO3(R). these functions will be denoted using bold characters, such as h : [0, T ] → R or R : [0, T ] → SO3(R). The same kind of notation will be used for three other time dependent vector fields k, ω, η and ξ which will be defined in the sequel. The five time dependent fields mentioned above will define the state z of the fluid-solid system. A vector from R or a matrix from SO3(R) will be denoted by h or by R, respectively. The transposed of a matrix will be denoted by ∗ so that the column vector of components a and b is denoted either ( a b ) or by (a, b)∗. Differentiation with respect to time is often denoted a dot. The vector, respectively the inner, product of v, w ∈ R will be denoted by v∧w and v · w, respectively . The Jacobian matrix of a vector field y 7→ f(y) defined on an open subset of R will be denoted by Dyf or simply by Df . ∗Insitut Élie Cartan, Nancy Universités/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lèsNancy Cedex, France, [email protected] †Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático , Universidad de Chile, Casilla 170/3 Correo 3, Santiago, Chile, [email protected] ‡Insitut Élie Cartan , Nancy Universités/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lèsNancy Cedex, France, [email protected]