Hasegawa-Mima equation in bounded domain with singular density

We investigate the well-posedness of Hasegawa-Mima equation (HME) in bounded domain with Dirichlet boundary condition and singular density under different regularity assumptions on the data. Our approach relies on the coupling of a fourth-order regularization of the HME, the spectral Galerkin method and an appropriate regularization of the singular density. Uniqueness holds in the class of solutions with very mild singularities à la Yudovich or with bounded densities.