Local existence and uniqueness of the solution to the 2D Hasegawa-Mima equation with periodic boundary conditions -

Abstract

Magnetic plasma confinement is one of the most promising ways in future energy production. To understand the phenomena related to energy production through plasma confinement, several mathematical models can be found in literature, of which the simplest and powerful 2D turbulent system model is the Hasegawa-Mima equation, which describes the time evolution of drift wave. As a cultural note, the Hasegawa-Mima equation is also referred as the Charney-Hasegawa-Mima equation in geophysical context. In this thesis, we give a new elementary construction of periodic Sobolev spaces on an open square domain on which we prove the local existence and uniqueness of the solution to the Hasegawa-Mima equation coupled with necessary periodic boundary conditions. We do this via perturbing it into a semilinear abstract Cauchy problem and using analytic semigroup methods to obtain a local solution. Employing some a priori estimates, we obtain a local solution to the original problem. We finally establish its uniqueness, and comment on the existence of a global solution.