AI 中文总结
研究在\(\mathbb{R}^2\)及光滑有界域中无粘表面准地转方程的全局弱解，针对临界洛伦兹空间\(L^{4/3,2}\)的任意初始数据。核心方法是用为洛伦兹空间定制的近似方案，主要贡献是构造出守恒哈密顿量的弱解。
AI 中文摘要
我们在\(\mathbb{R}^2\)和光滑有界区域中构造了无粘表面准地转方程的全局弱解，适用于临界洛伦兹空间\(L^{4/3,2}\)中的任意初始数据。这些解在所有时间都守恒哈密顿量\(\|\Lambda^{-1/2}\theta(t)\|_{L^2}^2\)。第二个洛伦兹指数由尖锐有界性\(\Lambda^{-1/2}:L^{4/3,2}\to L^2\)确定；当\(q>2\)时，对于\(L^{4/3,q}\)相应估计不成立。我们使用为洛伦兹空间量身定制的近似方案，它使平流速度和初始数据平滑，并在分布函数中保持阶数。通过解的高振幅截断的洛伦兹范数的一致界来去除近似。
英文摘要
We construct global weak solutions of the inviscid surface quasi-geostrophic equation in $\mathbb R^2$ and in smooth bounded domains, for arbitrary initial data in the critical Lorentz space $L^{4/3,2}$. The solutions conserve the Hamiltonian $\|Λ^{-1/2}θ(t)\|_{L^2}^2$ for all times. The second Lorentz exponent is determined by the sharp boundedness $Λ^{-1/2}:L^{4/3,2}\to L^2$; the corresponding estimate fails for $L^{4/3,q}$ when $q>2$. We use an approximation scheme that is tailored for Lorentz spaces. It smooths the advecting velocity and the initial data, and preserves order in distribution functions. Removing the approximation is made possible by uniform bounds for the Lorentz norms of high amplitude cutoffs of the solutions.
