AI 中文总结
该研究针对γ∈(0,1)的受迫广义SQG方程，构造了初始数据与力，证明其经典有限能量解在Sobolev适定性区域内于t=1时发生有限时间爆破，为该领域首个此类结果。
AI 中文摘要
我们针对奇异速度区域γ∈(0,1)下的受迫广义表面准地转方程，证明了其有限时间奇性形成结果，其中γ=0对应标准SQG方程。对于每个此类γ，我们构造了一个光滑、具有紧支集的初始数据和一个随时间变化的力F，使得对应的解θ在[0,1)上为经典解，且所有时刻均具有有限能量，但在t=1时失去Sobolev正则性。更准确地说，存在κ₀>2+γ+γ²(1−γ)/(25(4+γ))，使得对于每个κ∈[2+γ,κ₀]，力满足F∈L¹([0,1];Hᵏ(R²))，而对于同一区间内的每个指数，lim_{T↗1}∫₀ᵀ‖θ(·,t)‖_{Hᵏ}dt=∞。与此同时，对于任何κ₁∈[0,2+γ−γ(1−γ)/(2(4+γ))]，解在其整个寿命期内始终在Hᵏ¹中一致有界。由此可见，奇性发生在Sobolev适定性区域内，且不能归因于力或初始数据的正则性不足。据我们所知，这是广义SQG方程的经典有限能量解在适定性区域内的首个有限时间爆破结果。
英文摘要
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.
