Large comportement au temps large de l'equation de Prandtl et des systemes de magnetohydrodynamique
This thesis is made up of two parts. One is about the long time wellposedness of Prandtl equations with monotonicity assumption. The other one is the study of global solutions for inhomogeneous Magnetohydrodynamics system with bounded positive density. Recently, under the monotonic assumption, by using the energy method, Alexandre Wang Xu Yang and Masmoudi Wong have obtained the local in time existence of smooth solution in Sobolev space for Prandtl boundary layer equation, but the life span of their solution are very small. On the meantime, Xin Zhang proved the global in time weak solution by Crocco transformation under monotonicity and favorable pressure assumption. The long time behavior of the Prandtl equations is important to make progress towards the inviscid limit of the Navier Stokes equations. With this motivation, in the first part of this thesis, we study the long time well posedness for the nonlinear Prandtl boundary layer equation on the half plane. We consider a class of the initial data as perturbations around a monotonic shear profile and we prove the existence, uniqueness and stability of solutions in weighted Sobolev space, whose life span can be arbitrarily long while the initial perturbations are small enough. We use the energy method to prove the existence of solutions by a parabolic regularizing approximation. The nonlinear cancellation properties of Prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate. The second part of this thesis is about global well posedness of inhomogeneous magnetohydrodynamics(MHD) system. Recently, Danchin Mucha have obtained well posedness of inhomogeneous Navier Stokes equation while the density could be discontinuous by using Lagrangian transformation, or the material derivative. We will prove the global well posedness of inhomogeneous MHD system while the density just has a positive lower bound and the initial magnetic field contains large oscillations. We first get the a priori estimate in Euler coordinate and then prove the local in time well posedness of inhomogeneous MHD system in Lagrangian coordinate. Moreover, local solutions become global if the usual H1 norm of velocity and L2\L4 norm of magnetic field are small enough. Here, the smallness assumptions are different on initial velocities and initial magnetic fields. Moreover, we don't need to demand gradient of magnetic field to be small enough as that of velocities. So the initial magnetic filed can contain large oscillation.