AI 中文总结
该研究针对二维环面上的无粘表面准地转方程，利用凸积分格式在$p=4/3+10^{-5$的主动标量下，证明了弱解的灵活性与非唯一性，拓展了浓度临界指数相关结论。
AI 中文摘要
受文献[BCK26]启发，我们针对二维环面上的无粘表面准地转方程提出了一种新的凸积分格式。对于显式非最优指数$\bar p=\frac{4}{3}+10^{-5}$，我们在带有主动标量$\theta$的标准动量公式下证明了弱解的灵活性定理，其中$\theta \text{属于} C([0,1];L^{\bar p}(\text{mathbb} T^2))$。更确切地说，$L^{\bar p}(\text{mathbb} T^2)$中任意两个给定的均值为零的状态，都可以通过这样的解在初始时刻和最终时刻逼近。这些扰动由局部化、集中的行波SQG剖面构造而成，其中心沿有理方向移动，且半径依赖于雷诺应力。沿这些轨迹的辅助源的时间平均重构了前一阶段的应力，而二维双线性零形式估计则补偿了由非局部本构定律引起的导数损失。利用迭代的时间局部性，我们还得到了均值为零的空间$L^{\bar p}(\text{mathbb} T^2)$的一个稠密子集，使得该子集中的每个初始数据都至少允许两个不同的动量弱解。因此，该构造在浓度临界指数$p=4/3$之外同时建立了灵活性和非唯一性。
英文摘要
We develop a new convex-integration scheme, inspired by \cite{BCK26}, for the inviscid surface quasi-geostrophic equation on the two-dimensional torus. For the explicit, nonoptimized exponent $\bar p=\frac{4}{3}+10^{-5},$ we prove a flexibility theorem for weak solutions in the standard momentum formulation with active scalar \[ θ\in C([0,1];L^{\bar p}(\mathbb T^2)). \] More precisely, any two prescribed mean-zero states in $L^{\bar p}(\mathbb T^2)$ can be approximated at the initial and final times by such a solution. The perturbations are constructed from localized, concentrated traveling SQG profiles whose centers move along rational directions and whose radii depend on the Reynolds stress. Time averages of auxiliary sources along these trajectories reconstruct the preceding-stage stress, while a two-dimensional bilinear null-form estimate compensates for the derivative loss caused by the nonlocal constitutive law. Exploiting the time-locality of the iteration, we also obtain a dense subset of the mean-zero space $L^{\bar p}(\mathbb T^2)$ such that every initial datum in this subset admits at least two distinct momentum weak solutions. Thus, the construction establishes both flexibility and nonuniqueness beyond the concentration-critical exponent $p=4/3$.
