双曲平均曲率流

Hyperbolic mean curvature flow is an interdisciplinary study of the theory of hyperbolic partial differential equations and differential geometry, modern physics, crystallography, etc. It is one of the frontier key research topics and plays an important role in the study of geometry, modern physics, crystallography (in particular, the theory of melting crystals of helium) and the dynamics of motion of surfaces in the 3-dimensional Euclidean space. This project will investigate the following problems: .. (1) The global well-posedness theory of smooth solutions for the equations of hyperbolic mean curvature flow and the long-time behavior of global classical solutions; .. (2) The existence of time-periodic solutions and solutions of two point boundary value problems for the hyperbolic mean curvature flow; .. （3）The singularity theory of smooth solutions for hyperbolic mean curvature flow in which the typical problems include: when and where do the solutions blow up? what quantities blow up? how do they blow up? what kinds of singularities appear? how do the singularities grow out of nothing and propagate? .. (4) The applications of hyperbolic mean curvature flow to both geometry and physics: using the hyperbolic mean curvature flow, we study the geometry for the motion of surfaces in the 3-dimensional Euclidean space; using the hyperbolic mean curvature flow, we investigate the nonlinear dynamics of the solid-liquid interface in crystallography, in particular, we study the nonlinear dynamics of the solid-liquid interface in melting crystals of helium, since melting crystals of helium exhibit a phenomenon generally not found in other materials: oscillations of the solid-liquid interface in which atoms of the solid move only when they melt and enter the liquid. ..Clearly, the solution of these problems has great scientific significance in both theoretical and applied aspects.

双曲平均曲率流是将双曲型偏微分方程理论与微分几何学、现代物理学、晶体学等领域相交叉的前沿主流研究课题，它对几何学、现代物理学、晶体学以及曲面运动的非线性动力学等学科具有十分重要的意义。本项目将着重研究下述几个方面的问题：（1）双曲平均曲率流方程光滑解的整体适定性理论以及整体解的长时间性态；（2）时间周期解与两点边值问题解的存在性；（3）双曲平均曲率流方程光滑解的奇性理论：研究在什么条件下双曲平均曲率流方程的光滑解会在有限时刻内破裂，在此基础上进一步研究解的精确破裂时间、破裂点集的几何性质，重点研究奇性的形成机制、奇性的种类以及奇性的传播方式等问题；（4）双曲平均曲率流在几何、物理等学科中的应用：利用双曲平均曲率流，研究3维空间中曲面运动的几何学；研究氦晶体熔化或结晶时固-液界面的运动规律，刻画固-液界面运动的非线性动力学。这些问题的解决无论是在理论上还是在应用方面均具有十分重要的科学价值。

双曲平均曲率流是将双曲型偏微分方程理论与微分几何学、现代物理学、晶体学等领域相交叉的前沿主流研究课题，它对几何学、现代物理学、晶体学以及曲面运动的非线性动力学等学科具有十分重要的意义。本项目着重研究了下述几个方面的问题：（1）双曲平均曲率流方程光滑解的整体适定性理论以及整体解的长时间性态；（2）时间周期解与两点边值问题解的存在性；（3）双曲平均曲率流方程光滑解的奇性理论：研究在什么条件下双曲平均曲率流方程的光滑解会在有限时刻内破裂，在此基础上进一步研究解的精确破裂时间、破裂点集的几何性质，重点研究奇性的形成机制、奇性的种类以及奇性的传播方式等问题；（4）双曲平均曲率流在几何、物理等学科中的应用：利用双曲平均曲率流，研究3维空间中曲面运动的几何学；研究氦晶体熔化或结晶时固-液界面的运动规律，刻画固-液界面运动的非线性动力学。这些问题的解决无论是在理论上还是在应用方面均具有十分重要的科学价值。