二维 Euler 方程中 loglog 涡传播时连续模的保持与破坏
Conservation and Breakdown of Modulus of Continuity in the Propagation of Loglog Vortex for 2D Euler equation
AI总结：
本文研究二维 Euler 方程中 loglog 涡在连续扰动下的奇异结构持续性，证明 $0<α<1$ 时连续模可保持传播，并构造光滑扰动说明 $α>1$ 时范数瞬间爆破，从而确立阈值的尖锐性。
摘要：
本文研究在连续函数扰动下，二维 Euler 方程 loglog 奇异结构的持续性。首先，我们证明：当 loglog 涡受到连续模为 $μ(r)=\left(\log \frac{1}{r}\right)^{-α}$（$0<α<1$）的函数扰动时，loglog 涡能够传播，同时扰动项保持相同的连续模。随后，我们证明该结果是尖锐的：存在一个光滑函数，使得当 loglog 涡受到该光滑函数扰动时，对所有 $α>1$，对应连续模 $μ(r)=\left(\log \frac{1}{r}\right)^{-α}$ 的扰动项范数会瞬间爆破。
英文摘要：
In this paper, we investigate the persistency of loglog singular structure for 2D Euler equation under the perturbation of continuous function. First, we prove that when loglog vortex is perturbed by the function with modulus of continuity $μ(r)=\left( \log \frac{1}{r}\right)^{-α} ~~ (0<α<1)$, the loglog vortex is propagated while the perturbation term maintaining same modulus of continuity. We then show that this result is sharp: There exists a smooth function such that when the loglog vortex is perturbed by the smooth function, the norm of perturbation term corresponding to modulus of continuity $μ(r)=\left( \log \frac{1}{r}\right)^{-α} $ instantly blows up for all $α>1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
