This project aims at developing the study of three fundamental equations from physics. First, stability issues for the 2-fluids Euler-Maxwell equation, which is one of the fundamental equations in plasma physics, are considered. The goal is to prove that, under certain conditions, small perturbations of an equilibrium will not develop shocks, and that, actually, the plasma will get back to equilibrium, even in the absence of dissipation. Such a result would be of great physical and mathematical importance as it is known to be false for the compressible Euler equation in the absence of a self-consistent electromagnetic field. In a second part, a study is made of the Schrodinger equation on a curved background. In this case, many classical tools from the Euclidean theory break down and one expects the appearance of many new phenomena due to the influence of the geometry of the background. In particular, a study will be made of the effect of the growth of the volume on the global existence and global behavior of the solutions to the energy-critical equation. Finally, in a third part, ideas developed before are used to upgrade some known results about the homogeneous fourth-order equation to the more physical inhomogeneous equation.
Understanding how a fluid can be stabilized by a self-consistent electromagnetic field in the absence of any friction represents a cross-disciplinary collaboration between pure mathematics, applied mathematics, and physics, with applications in fluid engineering. More specifically, proving that a plasma at rest is stable under small perturbations would be a major physical discovery and would most certainly greatly enhance our ways to control plasma. This is especially fitting since plasma stability is one of the main factor limiting performances in tokamaks. Finally, beyond industrial applications, plasma represents the state of more than 99% of the matter in the universe and any work providing better understanding of this state would be of great importance. Dispersive equations and equations on curved spaces (manifolds) provide a rich area of interaction between various branches of mathematics as well as between different sciences. The study of equations on curved spaces is of interest to geometers, analysts, and number theorists in mathematics, as well as to theoretical physicists working in quantum chaos and general relativity. Fourth-order equations naturally arise in many different branches of physics and mathematics, especially those linked with elasticity and are critical to understand phenomena as diverse as the movement (and possible oscillations and breakdown) of bridges or the structure of blood vessels and the interaction between their membranes and the biofluids.
