SQG 的 Onsager 型定理
An Onsager-type theorem for SQG
AI总结：
本文利用 Nash 迭代与线性解耦方法构造时间紧支集的 SQG 非平凡弱解，证明其违反 Hamiltonian 守恒，从而在临界正则性意义上解决 SQG 的 Onsager 猜想。
摘要：
我们构造了表面准地转（SQG）方程的非平凡弱解 $θ\in C_t^0C_x^{0-}$，这些解在时间上具有紧支集，因此违反 Hamiltonian 守恒。鉴于这类守恒律对 $C_{t,x}^0 \subset L_{t,x}^3$ 类中的所有弱解均成立（Isett–Vicol，2015），该结果是临界的，并解决了 SQG 的 Onsager 猜想。该构造通过 Nash 迭代以及 Giri–Radu（2023）最近引入的线性解耦方法实现。
英文摘要：
We construct non-trivial weak solutions $θ\in C_t^0C_x^{0-}$ to the surface quasi-geostrophic (SQG) equations, which have compact support in time and, thus, violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class $C_{t,x}^0 \subset L_{t,x}^3$ (Isett-Vicol, 2015) and resolves the Onsager conjecture for SQG. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced in Giri-Radu (2023).
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