发表机构
University of Basel（巴塞尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对具粗糙初值的周期Benjamin--Ono方程，利用Gérard等人的光滑化定理分析其Talbot效应，得到规范-Hardy表示等严格结果，定性与相关数值结果一致。
AI 中文摘要
我们研究具有粗糙初值的周期Benjamin--Ono方程的Talbot效应。利用Gérard--Kappeler--Topalov针对Tao规范变换的光滑化定理，我们将奇异性分析简化为显式二次规范轮廓。对一般有界变差（BV）数据，我们得到有理时间的规范-Hardy表示；对一类自然的Talbot容许数据（其初始规范轮廓仅含有限个边奇异性），该表示可分解为有限个对数核与单跳跃核的和，外加一个连续余项。我们证明有限跳跃的分段光滑BV数据是Talbot容许的，特别地，这适用于方波。在无理时间下，我们在有限Diophantine型条件下证明连续性，同时展示了一个Liouville障碍，表明对应的单侧规范轮廓并非在所有无理时间都连续。这些结果给出了严格的结构结果，其定性与Alama Bronsard--Laurens的数值轮廓一致。
英文摘要
We study the dependence on time of the spatial regularity of solutions to the periodic Benjamin--Ono equation with real, mean-zero initial data of bounded variation. For every datum with a nonzero jump, we prove that continuity and essential boundedness at an irrational time $t=2πα$ are equivalent to the arithmetic condition $\sum_j\log(q_{j+1})/\sqrt{q_j}<\infty$, where $q_j$ are the continued-fraction denominators of $α$. We also determine the exact Hölder regularity and the Besov scale for spatial exponents $2<p\leq\infty$. We derive a high-frequency energy formula and prove the same sharp local criteria on every spatial interval at every irrational time for an open dense set of BV data. At rational times, piecewise smooth data give finite sums of logarithmic and jump kernels modulo a continuous remainder. For arbitrary BV data with a jump, the nonlinear and linear solutions have different logarithmic Poisson coefficients at every sufficiently large reduced denominator, uniformly in the numerator. The proofs combine the smoothing of Tao's gauge transform with estimates for quadratic Fourier series. A finite averaging argument isolates an atom of the initial gauge measure while retaining the real nonlinear reconstruction.
