Collaborative Reserach on Nonlinear PDEs and Integro-differential equations in the complex plane

NSF奖摘要- DMS-0103829数学科学：合作研究：复平面上的非线性偏微分方程和积分微分方程摘要DMS-0103829 Tanveer研究的重点是新的复分析方法，利用Borel-Laplace对偶来研究某些类别的非线性偏微分方程和积分微分方程的解的正则性和奇异性。复平面上的微分方程要研究的问题包括在物理应用中产生的方程，在某些临界值的参数是不适定的真实的域或结构不稳定。该方法使得能够控制接近参数空间中的临界点的敏感依赖。自然界中的许多现象都是由对参数敏感的方程来模拟的。这些方程出现在流体力学、晶体生长和图案形成的描述中。潜在的敏感性可能使这些问题难以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，以精确理解这些现象，我们的研究就是为了实现这一目标。

NSF Award Abstract - DMS-0103829Mathematical Sciences: Collaborative Research: Nonlinear partial differential equations and integro-differential equations in the complex planeAbstractDMS-0103829TanveerThe 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.