涡斑边界正则性保持的统一理论
Unified theory for regularity persistence of vortex patch boundaries
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中文总结 AI 辅助
针对含径向卷积核的二维有源标量方程，建立涡斑边界 Sobolev 正则性保持的统一局部理论，证明特定条件下轮廓动力学方程的局部解存在唯一性，丰富了这类方程的正则性研究成果。
中文摘要 AI 辅助
我们针对一类带有径向卷积核 $K(|x-y|)$ 的二维有源标量方程，建立了涡斑边界 Sobolev 正则性保持的统一局部理论。该类方程包含二维欧拉方程、局部可积范围 $0<\beta<1$ 内的广义 SQG 方程以及准地转浅水波方程。在 $K$ 的自然假设（光滑性、原点附近可积性、单调性及多项式增长）下，我们证明若初始边界属于 $H^3(\mathbb T)$ 且满足弧弦条件，则轮廓动力学方程在 $C([0,T];H^3(\mathbb T))$ 中存在唯一局部解；在核的更强可积性条件下，还可得到 $H^2$ 解的局部存在性。证明结合了轮廓方程的 Sobolev 能量估计与弧弦量的定量控制。
英文摘要
We establish a unified local theory for the persistence of Sobolev regularity of vortex patch boundaries in a family of two-dimensional active scalar equations with radial convolution kernels $K(|x-y|)$. The class includes the 2D Euler equation, the generalized SQG equation in the locally integrable range $0<β<1$, and the quasi-geostrophic shallow water equation. Under natural assumptions on $K$ (smoothness, integrability near the origin, monotonicity, and polynomial growth), we prove that if the initial boundary belongs to $H^3(\mathbb T)$ and satisfies the arc-chord condition, then the contour dynamics equation admits a unique local solution in $C([0,T];H^3(\mathbb T))$. Under a stronger integrability condition on the kernel, we also obtain local existence of $H^2$ solutions. The proof combines Sobolev energy estimates for the contour equation with quantitative control of the arc-chord quantity.
