Weak and dissipative solutions for the Hasegawa-Mima equation

We consider the Hasegawa-Mima equation in its ``Euler-like’’ velocity form: $${\mathrm{\partial }}_t(u-{\mathrm{\Delta }}^{-1}u)+(u\cdot \mathrm{\nabla })u-u^{\perp }\log n_0=0,$$ $n_0$ being the time-independent function appearing in the particle count $n=n_0{e}^{\frac{e\phi }{T}}$, and $u$ being the drift velocity ${\mathrm{\nabla }}^{\perp }\phi =-\mathrm{\nabla }\phi \times \widehat{z}$. 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.