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

We prove finite-time singularity formation for the forced generalized surface quasi-geostrophic equation in the singular velocity regime $γ\in(0,1)$, where $γ=0$ corresponds to SQG. For every such $γ$, we construct a smooth, compactly supported initial datum and a time-dependent force $F$ for which the corresponding solution $θ$ 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 $F\in L^1([0,1];H^κ(\mathbb{R}^2))$ for every $κ\in[2+γ,κ_0]$, 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 $H^{κ_1}$ 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.