Collaborative Research: Nonlinear PDEs and Integro-Differential Equations in the Complex Plane

NSF奖摘要-DMS-0103807数学科学：合作研究：复平面中的非线性偏微分方程组和积分-微分方程组摘要DMS-0103807成本研究重点是利用Borel-Laplace对偶来研究复平面中某些类型的非线性偏微分方程组和积分-微分方程解的正则和奇异性质的新的复分析方法。要研究的问题包括物理应用中出现的方程，这些方程在参数的某个临界值上要么在实域中不适定，要么在结构上不稳定。这些方法使得能够控制参数空间中接近临界点的敏感相关性。自然界中的许多现象都是由敏感地依赖于参数的方程模拟的。这样的方程出现在流体力学、晶体生长和图案形成的描述中。潜在的敏感性可能会使这类问题很难使用计算机模拟来解决。重要的是开发基于数学分析的理论工具，以使人们能够准确地理解这种现象，我们的研究就是朝着这个目标进行的。

NSF Award Abstract - DMS-0103807Mathematical Sciences: Collaborative Research: Nonlinear partial differential equations and integro-differential equations in the complex planeAbstractDMS-0103807CostinThe research focuses on new complex analytic methods exploiting Borel-Laplace duality to investigate regular and singular properties of solutions to certain classes of nonlinear partial differential equations and integro-differential equations in the complex plane. Problems to be studied include equations arising in physical applications that at some critical value of a parameter are either ill-posed in the real domain or are structurally unstable. The methods enable control over sensitive dependence close to a critical point in the parameter space. Many phenomena in nature are modeled by equations that depend sensitively on parameters. Such equations arise in descriptions of fluid mechanics, crystal growth and pattern formation. The underlying sensitivity can make such problems difficult to tackle using computer simulations. It is important to develop theoretical tools based on mathematical analysis to allow precise understanding of such phenomena, and our research is geared towards this goal.