Flexibility and rigidity in steady fluid motion

高维Navier-Stokes方程稳态解的刚性及其应用

Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications

本文证明高维（n≥4）Navier-Stokes方程满足|u(x)|≤C/|x|的稳态解必为零，通过加权能量估计给出初等简短证明，并应用于奇异性消除与无穷远渐近行为。

二维不可压缩Euler方程的刚性研究

On the rigidity of the 2D incompressible Euler equations

该研究针对二维有界区域定常Euler流的刚性问题，证明了圆盘、环域中满足切向边界条件的对应驻点条件下流必为环流，还揭示无滑移边界下的绝对刚性，验证了相关猜想并解决公开问题。

Boundary properties of fractional objects: flexibility of linear equations and rigidity of minimal graphs

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in~$\R^3$, i.e. nonlocal minimal surfaces with a graphical structure. We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum. This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can …

二维欧拉稳态的灵活性

On the flexibility of 2D Euler steady states

研究二维不可压缩欧拉方程的稳态解，发现非径向解析稳态解中流函数与涡度间存在全局关系，但光滑函数不满足此关系，甚至在结构假设下亦不成立，且存在多极点稳态可扰动为光滑稳态，涡度非单值函数。

环域中欧拉方程的一个刚性结果

A rigidity result for the Euler equations in an annulus

该研究针对二维有界环域定常欧拉流的刚性问题，通过分析流线几何性质与半线性椭圆方程解的局部对称性，证明满足无内驻点和切向边界条件的流动为环形流，解决了一则公开问题。

各向同性朗道方程的一个尖锐刚性/柔性阈值

A sharp rigidity/flexibility threshold for the isotropic Landau equation

该研究为各向同性朗道方程的Krieger–Strain方程建立了尖锐可积性阈值，用纳什迭代构造非平凡解，分离出磨光子汇合概念，是纳什迭代首次用于碰撞动理学非线性方程。

谱Ricci下界和均凸边界下的刚性与灵活性

Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary

研究具有均凸边界的Riemann流形，其谱Ricci曲率非负，证明在特定γ范围内，流形必须分解为积结构，并给出关于相对基本群的拓扑刚性结果。

平面中欧拉方程剪切流的刚性

Rigidity of shear flows of the Euler equations in the plane

本文研究了欧拉方程中剪切流的刚性问题，通过结合退化Monge-Ampère方程和传输方程理论，证明了特定条件下的刚性与灵活性差异。

二维外部区域中欧拉流的刚性结果

Rigidity results for Euler flows in two dimensional exterior domains

本文研究二维外部区域中不可压缩欧拉方程的稳态解，在边界入流及无穷远速度条件下，证明无旋流动等价于无驻点。

稳态层流的几何刻画

A geometric characterization of steady laminar flow

本文刻画周期渠道、环域及一般二连通区域中的稳态层流，证明其必为平行或圆周流，并揭示 Poiseuille 流邻域刚性的尖锐性。

Shear flows of an ideal fluid and elliptic equations in unbounded domains

We prove that, in a two-dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half-plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n-dimensional slabs in any dimension n.

Non-conservative solutions of the Euler-$α$ equations

The Euler-$α$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $α$. We show that there exists $β>1$ such that weak solutions to the two and three dimensional Euler-$α$ equations in the class $C^0_t H^β_x$ are not unique and may not conserve the Hamiltonian of the system, thus demonstrating flexibility in this regularity class. The construction utilizes a Nash-style intermittent convex integration scheme. We also formulate an appropriate version of the Onsager conjecture for Euler-$α$, postulating that the threshold between rig…

具有超二次增长的薄障碍问题的全局解

Global solutions to the thin obstacle problem with superquadratic growth

本文研究了具有超二次增长的薄障碍问题全局解的刚性/柔性，根据无穷远展开对有界正集解进行分类，并证明有界接触集解可逼近任意紧集，揭示了经典障碍问题中不存在的现象。

Convex integration for the Monge-Ampère equation in two dimensions

This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration à la Nash & Kuiper and establish the related h-principle for the Monge-Ampère equation in two dimensions

关于有界区域上 Arnold 第二稳定性定理中正常数的研究

On the positive constant in Arnold's second stability theorem for a bounded domain

本文研究有界区域上 Arnold 第二稳定性定理中的正常数，证明其可取为某类 Laplace 特征值问题的第一特征值，并探讨了达到该值时的失稳现象及圆盘中定常 Euler 流的刚性与轨道稳定性。