Liouville equations, limits in large number of particles.

Summary This thesis is devoted to ordinary differential equations and associated transport equations for sparsely regular vector fields, and contains four works. The first one deals with the solution of ODEs and transport equations for vector fields in L^2(R^2) with zero divergence, verifying a regularity condition on the field direction. The results are obtained in the framework of the theory developed by R. DiPerna and P. L. Lions for the solution of transport equations with sparse regular coefficients. We lower the necessary regularity conditions in this particular case of dimension two. The second work concerns the Liouville equation, which governs the behavior of a density of N interacting particles, in the framework of weakly regular fields. The results of DiPerna and Lions, already extended to the kinetic case by François Bouchut, are adapted to take into account a singularity at the origin. In the third work, we are interested in the convergence of interacting particle systems to the Vlasov equation. The convergence is obtained through accurate discrete estimates, in the case of 1/|x|^alpha interaction forces, for alpha < 1. This improves the previously known result for C^1 forces. The fourth one uses the same kind of technique for the Euler approximation by vortices. Convergence is proved for any time when the interaction is slightly less singular than for vortices. Uniform bounds on the field and its accretion in the case of real wormholes are also given.