Nonlinear waves in general relativity and fluid dynamics

Abstract

This thesis deals with the analysis of partial differential equations describing nonlinear wave-like phenomena in three different settings: general relativity, the compressible Navier--Stokes equations, and magnetohydrodynamics.

Many results on the global dynamics of hyperbolic equations, particularly in the case of the Einstein equations, rely on the use of the vector field method. This method requires the initial data to be highly localized around a single point in space. In this first part of the dissertation, we extend the classical vector field method of Klainerman to deal with initial data localized around several points whose pairwise distances are assumed to be large. We are therefore able to prove global stability for solutions to quasilinear wave equations satisfying the null condition when the initial data are not required to be localized around a single point. This probes a regime which was not accessible by previous physical-space methods. This part is based on joint work with John Anderson.

The second chapter of this thesis deals with the global dynamics of the compressible Navier—Stokes equations in one and two space dimensions. A particular case of these equations arises in geophysical fluid dynamics as the viscous shallow water equations. Concerning the one-dimensional model, we introduce a quantity, called the active potential, which allows us to control the dynamics for a large range of pressure and viscosity laws. As a byproduct, we are able to prove a conjecture formulated in 1994 by Peter Constantin. This part is based on joint work with Peter Constantin, Theodore Drivas, and Huy Nguyen.

Finally, the third chapter deals with the equations of magnetohydrodynamics (MHD). We prove that a suitably regularized Voigt—MHD model admits a global-in-time solution, which moreover converges, in the infinite time limit, to a solution of the steady three-dimensional incompressible Euler equations. This can be regarded as a rigorous construction of an MHD equilibrium by means of a process known in the physics literature as magnetic relaxation: the magnetic field is drawn towards equilibrium by an MHD-type system. This part is based on joint work with Peter Constantin.