Asymptotics and simulation of singular phenomena

非线性动力系统的定性研究对于理解科学技术中的许多现象至关重要。在燃烧理论、非线性光学、等离子体物理、流体力学、数学生物学和微分几何等许多不同领域出现的模型问题中，对这种行为的研究受到了广泛关注。在这些系统中，非线性反应项的影响可能克服空间扩散导致奇点的形成，其中解在有限时间内在某些范数内变得无界。这些奇点可能表明模型故障，因为在未捕获的尺度上存在重要的动态行为(例如。极度聚焦)或者它们可能对应于物理状态的质变(例如：爆炸)。仔细解决这种奇异行为附近的时空动力学对于理解许多纯和应用偏微分方程问题至关重要。我的研究涉及发展渐近和数值方法来研究这种微妙的现象。具体而言，项目包括：1)物理和微分几何中非线性偏微分方程的有限时间奇异性分析；2)多维移动网格方法的分析与实现；3)爆破问题的尺度不变运动网格方法分析与实现。

The qualitative study of nonlinear dynamical systems is crucial in understanding many phenomena in science and technology. The study of such behaviour has received much attention in model problems arising from numerous diverse areas including combustion theory, nonlinear optics, plasma physics, hydrodamics, mathematical biology and differential geometry. In these systems the effect of nonlinear reaction terms may overcome spatial diffusion leading to singularity formation wherein the solution becomes unbounded in some norm at a finite time.  These singularities may indicate model breakdown where there is important dynamic behaviour on scales not captured (eg. extreme focussing) or they may correspond to a qualitative change in the physical state (eg. explosion).  A careful resolution of the spatio-temporal dynamics near such singular behaviour is crucial to understanding many pure and applied PDE problems.  My research involves developing asymptotic and numerical approaches to studying this delicate phenomenon. Specifically, the project includes: 1) Analysis of finite-time singularities in nonlinear PDEs arising in physics and differential geometry; 2) Analysis and implementation of moving mesh methods in multiple dimensions; 3) Analysis and implementation of scale-invariant moving mesh methods for blow-up problems.