发表机构
Max-Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明二维正方形中Shnirelman不等式的一个版本，构造散度自由速度场连接目标流体构型与恒等映射，其范数由$L^p$差界定，因二维拓扑障碍允许轨迹相交，导致不可逆行为。
AI 中文摘要
本文证明了二维正方形$[0,1]^2$中Shnirelman不等式的一个版本。更精确地说，给定目标流体构型$f$（一个保体积微分同胚），我们构造一个散度自由的速度场$v\in L^q_tL^p_x$，其流将$f$连接到恒等映射，且其$L^q_{t} L^p_x$-范数可有效地由$f$与$id$的$L^p$-差界定。然而，向量场的更高正则性可能受到影响。与高维情形的主要区别在于，由于维度$\nu=2$的拓扑障碍，我们必须允许不同的流体轨迹在空间中“相交”。我们还观察到，这一选择导致了欧拉动力学中不可逆行为的出现。
英文摘要
In this paper we prove a version of Shnirelman's inequality in the two-dimensional square $[0,1]^2$. More precisely, given a target fluid configuration $f$ (a volume-preserving diffeomorphism) we construct a divergence-free velocity field $v\in L^q_tL^p_x$ whose flow connects $f$ to the identity and whose $L^q_{t} L^p_x$-norm can effectively be bounded by the $L^p$-difference of $f$ and $id$. However, higher regularity of the vector field might be affected. The main difference with the higher dimensional case is that, because of the topological obstructions of dimension $ν=2$, we have to allow that different fluid trajectories 'intersect' in space. We also observe that this choice leads to the emergence of irreversible behaviors in the Eulerian dynamics.
