Weak and dissipative solutions for the Hasegawa-Mima equation

Abstract

We consider the Hasegawa-Mima equation in its ``Euler-like'' velocity form: \[\partial_t(u-Δ^{-1}u)+(u\cdot\nabla)u-u^\perp\log n_0=0,\] $n_0$ being the time-independent function appearing in the particle count $n=n_0{e}^{\frac{e\phi }{T}}$, and $u$ being the drift velocity ${\nabla }^{\perp }\phi =-\nabla \phi \times \hat{z}$. Adapting the notion from Lions' book on the Euler equations, we prove the existence of dissipative solutions for this equation for any $L^2$ divergence free initial condition $w\in L^2(D)$, for $D={\mathbb{T}}^2$ and $D\subset {\mathbb{R}}^2$ a bounded ${\mathcal{C}}^1$ domain.