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

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.