Logarithmic singularities of solutions to nonlinear partial differential equations

具有部分对称性的对数能量超临界非线性波动方程的全局适定性

Global well-posedness for the logarithmically energy-supercritical Nonlinear Wave Equation with partial symmetry

本文研究部分对称初值下对数能量超临界非线性波动方程，利用适配对称的加权Morawetz与Strichartz估计，建立全局适定性与散射结果，推广了Tao的径向对称情形工作，并给出能量临界问题的定量结果。

三维纳维-斯托克斯方程中涡旋拉伸的对数耗散与奇点规避

Logarithmic Depletion of Vortex Stretching and Singularity Evasion in the 3D Navier-Stokes Equations

研究三维不可压缩纳维-斯托克斯方程临界点奇点，提出几何分析机制，通过对数加权空间及相关估计等，使涡度拉伸耗散，将对数增益转移到速度场，避免有限时间爆破。

欧氏空间薛定谔算子在对数区域的全纯函数零点与共振分布

Distribution of zeros of holomorphic functions and resonances in logarithmic regions for Schroedinger operators on Euclidean space

本文针对全纯函数零点分布建立复分析结果，将其应用于奇数维欧氏空间薛定谔算子，证明一维类跳跃奇点势可产生大量甚至无穷多串共振，并关联其性质与势参数及散射行列式渐近行为。

Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map

We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{ö}dinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp. L2) potentials whose difference is lying in any Sobolev space of order positive order.

Anisotropic singularities to semilienar elliptic equations in a measure framework

The purpose of this article is to study very weak solutions of elliptic equation $$ -Δu+g(u)=2k\frac{\partial δ_0}{\partial x_N }+jδ_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $δ_0$ is the Dirac mass concentrated at the origin and $\frac{\partialδ_0}{\partial x_N}$ is defined in the distribution sense that $$ \langle\frac{\partial δ_0}{\partial x_N},ζ\rangle=\frac{\partialζ(…

障碍问题中自由边界正则部分在奇点附近的结构

The structure of the regular part of the free boundary close to singularities in the obstacle problem

本文证明了障碍问题自由边界正则部分在奇点附近精细渐近行为的首个结果，源于对长期猜想的部分解答和高维无界接触集全局解的部分分类。

部分约束下对数薛定谔方程的基态与尖锐对数索伯列夫不等式

Ground states and sharp logarithmic Sobolev inequalities for logarithmic Schrödinger equations with partial confinement

本文针对带部分约束的对数薛定谔方程，建立抽象变分原理，证明其基态存在性，推导尖锐对数索伯列夫不等式并确定最优常数，还分类了不同情形下的基态。

Holder estimates for the $\bar\partial$-equation on surfaces with simple singularities

Let $Σ$ be a 2-dimensional subvariety in $C^3$ with an isolated simple (rational double point) singularity at the origin. The main objective of this paper is to solve the $\dbar$-equation on a neighbourhood of the origin in $Σ$, demanding a Holder condition on the solution.

Existence, Uniqueness and Removable Singularities for Nonlinear Partial Differential Equations in Geometry

This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.

Remarks on the Hölder-continuity of solutions to parabolic equations with conic singularities

This is a note on \cite{LSU} and \cite{FS}. Using their work line by line, we prove the Hölder-continuity of solutions to linear parabolic equations of mixed type, assuming the coefficient of $\frac{\partial}{\partial t}$ has time-derivative bounded from above. On a Kähler manifold, this Hölder estimate works when the metrics possess conic singularities along a normal crossing divisor.

在对数格点上追踪流体的复奇点

Tracking complex singularities of fluids on log-lattices

本文在对数格点流体动力学中研究Navier-Stokes方程复奇点猜想，利用奇点条带法追踪1D和3D模型中的主导复奇点，揭示了趋近实轴的新标度关系及不同耗散机制的影响。

通过部分数据的辐射传输方程数值恢复源奇异性

Numerical Recovery of Source Singularities via the Radiative Transfer Equation with Partial Data

本文针对辐射传输方程的部分数据逆源问题，数值计算法算子X_V^*X_V，并利用截断Neumann级数近似，有效恢复内部光学源f的可见奇异性。

Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$\partial_tU+ H(x,\partial_xU)=0,$$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $Σ(U)$, of points where $U$ is not differentiable, is locally contractible. Moreover, we study the homotopy type of $Σ(U)$. We also give an application to the singularities of a distance function to a closed subset of a complete Riemannian manifold.

Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises

We study the asymptotic properties of the stochastic Cahn-Hilliard equation with the logarithmic free energy by establishing different dimension-free Harnack inequalities according to various kinds of noises. The main characteristics of this equation are the singularities of the logarithmic free energy at 1 and --1 and the conservation of the mass of the solution in its spatial variable. Both the space-time colored noise and the space-time white noise are considered. For the highly degenerate space-time colored noise, the asymptotic log-Harnack inequality is established under the so-called ess…