A remark on the geometric interpretation of the A3w condition from optimal transport

Remarks on geometric properties of SQG sharp fronts and $α$-patches

Guided by numerical simulations, we present the proof of two results concerning the behaviour of SQG sharp fronts and $α$-patches. We establish that ellipses are not rotational solutions and we prove that initially convex interfaces may lose this property in finite time.

二维粗糙向量场消失李括号的几何解释

Geometric interpretation of the vanishing Lie Bracket for two-dimensional rough vector fields

本文证明了实平面上具有有界散度的连续 Sobolev 向量场在零李括号条件下其流可交换，从而去除了 Colombo-Tione（2021）所需的弱李可微性附加假设，并探讨了向 BV 情形的推广。

Elliptic Integrable Systems: a Comprehensive Geometric Interpretation

We give a geometric interpretation of all the $m$-th elliptic integrable systems associated to a $k'$-symmetric space $N=G/G_0$ (in the sense of C.L. Terng). It turns out that we have to introduce the integer $m_{k'}$ defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the primitive case ($m < m_{k'}$), the determined case ($m_{k'}\leq m \leq k'-1$) and the underdetermined case ($m \geq k'$). We prove that we have an interpretation in terms of a sigma model with a Wess-Zumino term. Moreover we prove that we have a geometric interpretation in terms of t…

Remarks on global existence of classical solution to multi-dimensional compressible Euler-Poisson equations with geometrical symmetry

We give a necessary and sufficient condition for the global existence of the classical solution to the Cauchy problem of the compressible Euler-Poisson equations with radial symmetry. We introduce a new quantity which describes the balance between the initial velocity of the flow and the strength of the force governed by Poisson equation.