Blow-up at finite time for the generalized SQG equations in the Sobolev well-posedness regime

Abstract

We prove finite-time singularity formation for the forced generalized surface quasi-geostrophic equation in the singular velocity regime $\gamma \in (0,1)$, where $\gamma =0$ corresponds to SQG. For every such $\gamma$, we construct a smooth, compactly supported initial datum and a time-dependent force $F$ for which the corresponding solution $\theta$ is classical on $[0,1)$ with finite energy for all times, but loses Sobolev regularity at $t=1$. More precisely, there exists \[ κ_0>2+γ+\frac{γ^2(1-γ)}{25(4+γ)}, \] such that the force satisfies $F\in L^1([0,1];H^{\kappa }({\mathbb{R}}^2))$ for every $\kappa \in [2+\gamma ,{\kappa }_0]$, whereas \[ \lim_{T\nearrow1}\int_0^T\|θ(\cdot,t)\|_{H^κ}\,dt=\infty \] for every exponent in the same interval. At the same time, the solution remains uniformly bounded in $H^{{\kappa }_1}$ throughout its lifespan for any \[κ_1\in\left[0,2+γ-\frac{γ(1-γ)}{2(4+γ)}\right].\] Then, the singularity occurs within the Sobolev well-posedness regime and cannot be attributed to insufficient regularity of the force or the initial conditions. To the best of our knowledge, this is the first finite-time blow-up result for classical finite-energy solutions of the generalized SQG equations in a well-posedness regime.