Weak and dissipative solutions for the Hasegawa-Mima equation

We consider the Hasegawa-Mima equation in its ``Euler-like'' velocity form: \[\partial_t(u-\Delta^{-1}u)+(u\cdot\nabla)u-u^\perp\log n_0=0,\] $n_0$ being the time-independent function appearing in the particle count $n=n_0e^{\frac{e\varphi}{T}}$, and $u$ being the drift velocity $\nabla^\perp\varphi=-\nabla\varphi\times\hat z$.

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.