Title:

Some problems in the mathematical theory of fluid mechanics

This thesis addresses three problems related to the mathematical theory of fluid mechanics. Firstly, we consider the threedimensional incompressible NavierStokes equations with an initial condition that has H1Sobolev regularity. We show that there is an a posteriori condition that, if satisfied by the numerical solutions of the equations, guarantees the existence of a strong solution and therefore the validity of the numerical computations. This is an extension of a similar result proved by Chernyshenko, Constantin, Robinson & Titi (2007) to less regular solutions not considered by them. In the second part, we give a simple proof of uniqueness of fluid particle trajectories corresponding to the solution of the ddimensional Navier Stokes equations, d = 2, 3, with an initial condition that has H(d/2)−1Sobolev regularity. This result has been proved by Chemin & Lerner (1995) using the LittlewoodPayley theory for the flow in the whole space Rd. We provide a significantly simpler proof, based on the decay of Sobolev norms ( of order more than (d/2)−1 ) of the velocity field after the initial time, that is also valid for the more physically relevant case of bounded domains. The last problem we study is the motion of a fluidrigid disk system in the whole plane at the zero limit of the rigid body radius. We consider one rigid disk moving with the fluid flow and show that when the radius of the disk goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the NavierStokes equations describing the motion of only fluid in the whole plane. We then prove that the trajectory of the centre of the disk, at the zero limit of its radius, coincides with a fluid particle trajectory. We also show an equivalent result for the limiting motion of a spherical tracer in R3, over a small enough time interval.
