Asymptotics and simulation of singular phenomena

非线性动力系统的定性研究对于理解科学和技术中的许多现象至关重要。这种行为的研究受到了广泛的关注，在模型问题所产生的许多不同的领域，包括燃烧理论，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。在这些系统中，非线性反应项的影响可以克服空间扩散，导致奇异性的形成，其中的解决方案在有限的时间变得无界的一些规范。 这些奇异性可能表明模型故障，其中在未捕获的尺度上存在重要的动态行为（例如，极端聚焦）或者它们可能对应于物理状态的质的变化（例如，爆炸）。 这种奇异行为附近的时空动力学的仔细决议是至关重要的理解许多纯粹的和应用的PDE问题。 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，该项目包括：1）物理和微分几何中出现的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和实现; 3）尺度不变移动网格方法的分析和实现爆破问题。

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.