Existence of weak solutions to the generalized SQG equations in Sobolev spaces

We consider the two-dimensional dissipative generalized surface quasi-geostrophic equations given by $${\mathrm{\partial }}_t\theta +u\cdot \mathrm{\nabla }\theta +\nu (-\mathrm{\Delta }{)}^{\alpha }\theta =0,u={\mathrm{\nabla }}^{\perp }{\mathrm{\Lambda }}^{\beta -2}\theta$$ and establish global existence of weak solutions with initial data in ${\dot{H}}^{\frac{\beta }{2}-1}({\mathbb{R}}^2)$. Our result extends the framework of Marchand’s solutions (Comm. Math. Phys. 277 (2008), no. 1, 45-67) to this larger family of active scalar equations.