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

Abstract: We consider the two-dimensional dissipative generalized surface quasi-geostrophic equations given by [\partial_t θ+u\cdot \nabla θ+ν(-Δ)^αθ=0,\quad u=\nabla^{\perp}Λ^{β-2}θ] and establish global existence of weak solutions with initial data in $\dot{H}<sup>{\fracβ{2}-1}(\mathbb{R}<sup>2)$. 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.