Strong and Weak Solutions to the Hasegawa-Mima Equation with Periodic Boundary Conditions

TL;DR

This paper introduces a new decoupling approach to the Hasegawa-Mima equation, reformulating it as a linear system to establish existence and uniqueness of weak and strong solutions under periodic boundary conditions.

Contribution

The paper develops a novel decoupling method transforming the HM equation into a linear PDE system, enabling the derivation of variational frameworks for weak and strong solutions with lower regularity assumptions.

Findings

Proved existence of global weak solutions for initial data in H_P^2 with W_0 in L^2.

Established local existence and uniqueness of strong solutions for higher regularity initial data.

Used Petrov-Galerkin systems and Fourier basis for constructing solutions.

Abstract

The two dimensional Hasegawa-Mima (HM) equation $$ -\Delta u_t+u_t = \{u,\Delta u\} + ku_y$$ describes the time evolution of drift waves in magnetically-confined plasma. Several authors have treated the HM equation theoretically and numerically, with difficulties arising when handling the non-linear Poisson's bracket $\{u,\Delta u\}:=u_x\Delta u_u-u_y\Delta u_x $. In this paper, we introduce a new decoupling approach that avoids the Poisson's bracket term by reformulating the HM equation as a system of two linear PDEs, a solution of which is a pair $(u,w)$ such that $$(HM)\,\,\,\left\{\begin{array}{lll} w_t + \vec{V}(u) \cdot \nabla w = ku_y\\ -\Delta u+u=w, \\ \end{array}\right.$$ where $\vec{V}(u)= -u_y \vec{\textbf{i}} + u_x \vec{\textbf{j}}$ is a divergence-free vector field. Based on this coupled hyperbolic-elliptic system, we derive several variational frames, all propitious for finding weak solutions with spacial periodic boundary conditions and lower regularity assumptions on the initial data. More precisely, for initial data $u_0 \in H_P^2(\Omega)$ with $w_0:=(I-\Delta) u_0 \in L^2(\Omega)$, we prove the existence of a weak solution that is global in time. And for initial data $u_0 \in H_P^3(\Omega)$ with $w_0:=(I-\Delta) u_0 \in H_P^1(\Omega) \cap L^\infty(\Omega)$, we prove the existence of a unique strong solution that is local in time. Our proofs are based on the existence of fixed-point ordered pairs $\{u_N,w_N\}$ that solve Petrov-Galerkin HM systems, constructed using spacial Fourier basis. Through appropriate a-priori estimates combined with compactness arguments, we reach when $N\to\infty$ limit point solutions $(u,w)$ to the (HM) system.