发表机构
Institute of Applied Physics and Computational Mathematics; School of Mathematics and Physics, University of Science and Technology Beijing（应用物理与计算数学研究所; 北京科技大学数理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对R²上的无粘表面准地转方程，证明了其Sobolev正则性在固定间隙下的瞬时损失，构造了满足特定正则性条件的初始数据并得到唯一解，明确了正则性损失的阈值关系。
AI 中文摘要
我们证明了R²上的无粘表面准地转方程在固定间隙下的Sobolev正则性的瞬时损失。更准确地说，对于每一个1<s<2、T>0和ε>0，我们构造初始数据θ₀∈Hˢ(R²)且||θ₀||_{Hˢ}≤ε，该数据在[0,T]上生成解θ，满足θ∈L^∞([0,T];H^{1+γ}(R²))，且对于每个t∈(0,T]、σ>σ₊(s)，θ(t)∉H^σ(R²)，其中γ>0和σ₊(s)满足1+γ<σ₊(s):=s(3-s)/(1+2s-s²)<s。该解在具有初始数据θ₀的确定经典解族中是唯一的。
英文摘要
We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on $\mathbb R^2$. More precisely, for every $1<s<2$, $T>0$ and $\varepsilon>0$, we construct initial data $θ_0\in H^s(\mathbb R^2)$ with $\|θ_0\|_{H^s}\leq\varepsilon$ that generate a solution $θ$ on $[0,T]$ such that \[ θ\in L^\infty([0,T];H^{1+γ}(\mathbb R^2)), \quad θ(t)\notin H^σ(\mathbb R^2) \quad\text{for every }t\in(0,T],\ σ>σ_*(s), \] where $γ>0$ and $σ_*(s)$ satisfy \[ 1+γ<σ_*(s):= \frac{s(3-s)}{1+2s-s^2}<s. \] The solution is unique in a determined family of classical solutions with initial datum $θ_0$.
Comments48 pages
