Spatially Inhomogeneous Bernstein's Problem and De Giorgi's Conjecture

具有有界指数的相变：与De Giorgi猜想的类比

Phase transitions with bounded index: Parallels to De Giorgi's conjecture

本文研究了具有有界指数的相变问题，证明了在高维空间中相变解的刚性行为，并与De Giorgi猜想进行了类比。

非局部算子动理论中的定量 De Giorgi 方法

Quantitative De Giorgi methods in kinetic theory for non-local operators

本文基于轨迹和能量估计中的非局部项，定量推导了动理学积分微分方程的 Harnack 不等式，蕴含 Hölder 连续性，并恢复了非截断情形下非齐次 Boltzmann 方程的已有结果。

Modica type gradient estimates for reaction-diffusion equations and a parabolic counterpart of a conjecture of De Giorgi

We continue the study of Modica type gradient estimates for non-homogeneous parabolic equations initiated in \cite{BG}. First, we show that for the parabolic minimal surface equation with a semilinear force term if a certain gradient estimate is satisfied at $t=0$, then it holds for all later times $t>0$. We then establish analogous results for reaction-diffusion equations such as \eqref{e0} below in $\Om \times [0, T]$, where $\Om$ is an epigraph such that the mean curvature of $\partial \Om$ is nonnegative. We then turn our attention to settings where such gradient estimates are valid withou…

Propagation of uniform boundedness of entropy and inhomogeneous regularities for viscous and heat conductive gases with far field vacuum in three dimensions

Due to the highly degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behave singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It is shown in this paper that the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities of the velocity and temperature can be propagated for viscous and heat conductive gases in $\mathbb R^3$, provided that the initial vacuum occurs only at far fields with suitably slow decay of the initial density. Precisely, it is proved that for any …

平均曲率流的平移解的伯恩斯坦问题

A Bernstein problem for translating solutions to the mean curvature flow

该研究针对平均曲率流的平移解的伯恩斯坦问题，在N≥8时构造出非凸的平均凸整体图平移子，完善了相关刚性理论。

Global existence of weak solutions to the Navier-Stokes equations with temperature-depending viscosity coefficient

In this paper, the initial-boundary value problem to the three-dimensional inhomogeneous, incompressible and heat-conducting Navier-Stokes equations with temperature-depending viscosity coefficient is considered in a bounded domain. The viscosity coefficient is degenerate and may vanish in the region of absolutely zero temperature. Global existence of weak solutions to such a system is established for the large initial data. The proof is based on a three-level approximate scheme, the De Giorgi's method and compactness arguments.