Long term regularity of some periodic nonlinear dispersive equations

Abstract

This thesis considers two periodic nonlinear dispersive equations arising from the study of fluid mechanics: (i) the 2D one-fluid Euler--Poisson system and (ii) the 3D gravity water wave system.

The one-fluid Euler--Poisson system is a basic plasma physics model, in which a compressible electron fluid flows under its own electrostatic field. The first part of the thesis concerns its periodic solutions, namely those living on the square torus. We prove long-term regularity of periodic solutions of this system in 2 spatial dimensions with small data. Our main conclusion is that on a square torus of side length $R$, if the initial data is sufficiently close to a constant solution, then the solution is wellposed for a time at least $R/(\epsilon^2(\log R)^{O(1)})$, where $\epsilon$ is the size of the initial data.

The gravity water wave system describes the motion of an incompressible fluid delineated from above by an interface, for exanple, the water in the ocean. The second part of the thesis studies its solutions periodic in the two horizontal directions, which describe a column of water whose base is a 2D torus. We also prove long-term regularity of periodic solutions of this system with small data.

Our main conclusion is that on a square torus of side length $R$, if the initial data is sufficiently close to a constant solution,

then the solution is wellposed for a time at least $R/(\epsilon^2(\log R)^{O(1)})$, where $\epsilon$ is the size of the initial data.