Colloquium / Seminars

Topic：The DiPerna-Majda 2D Gap Problem
Speaker：Prof. Daniel Spector
　　　　( National Taiwan Normal University)
Date time：Oct. 15, 2024 14:00 - 15:00
Venue：SA223
Abstract：
Abstract: In a series of influential papers in the 1980s, DiPerna and Majda introduced a rigorous framework of approximate solutions of the Euler equations and proved several results concerning concentration of solutions related to hypothesis on $\omega:= \operatorname*{curl}u:\mathbb{R}^2 \times [0, T] \to \mathbb{R}$ the \emph{vorticity} and $\omega_0:= \operatorname*{curl}u_0:\mathbb{R}^2 \to \mathbb{R}$ the \emph{initial vorticity}. Briefly, they proved that for vorticities bounded in an $\alpha$ log-Morrey space one does not have any concentration for $\alpha>1$, while for $\alpha\leq 1/2$ one may have \emph{concentration-cancellation}. The interval $\alpha \in (1/2,1]$ remained an open question in their paper and subsequent papers, whether one can rule out concentration or find sequences which admit concentration-cancellation. In this talk I discuss a recent result in collaboration with Oscar Dominguez in which we resolve this question, closing the gap, showing in particular that one may have concentration-cancellation up to $\alpha=1$.
Download：Talk_1131015.pdf

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