Large Solutions to Systems of Nonlinear Equations

数学科学：非线性方程组的大解[摘要]jenssen本研究的对象有三个方面。首先，我们寻求更好地理解双曲（无粘）守恒定律系统的解，这些系统的振幅或变化都很大。将考虑爆炸性行为的新例子，以及防止奇异行为的条件。接下来，我们将考虑可压缩流体的多维Navier-Stokes方程，并建立具有球形或圆柱对称的大解的整体存在性。最后，我们将考虑描述燃烧的流动。这是由Navier-Stokes方程建模的，并通过描述化学过程的方程进行扩充。在这种情况下，我们对波形的稳定性特别感兴趣。这项工作涉及非线性偏微分方程解的数学分析。研究将探讨守恒定律系统、可压缩流体流动和描述反应流动的方程。这种非线性方程的许多现有理论只适用于小解。然而，在气体流动、燃烧和爆炸等应用中，大溶液具有很大的兴趣，而这些溶液的研究需要在本项目中开发的新技术。

NSF Award Abstract - DMS-0206631Mathematical Sciences: Large Solutions to Systems of Nonlinear EquationsAbstract0206631 JenssenThe object of this research is threefold. First, we seek a better understanding of solutions to systems of hyperbolic (inviscid) conservation laws that are large in either amplitude or variation. New examples of explosive behavior will be considered, as well as conditions preventing singular behavior. Next, we will consider the multi-dimensional Navier-Stokes equations for a compressible fluid and establish global existence of large solutions with spherical or cylindrical symmetry. Finally, we will consider flow describing combustion. This is modeled by the Navier-Stokes equations augmented by equations describing the chemical processes. In this case we are particularly interested in the stability of wave patterns.The work deals with mathematical analysis of solutions to nonlinear partial differential equations. The research will investigate systems of conservation laws, compressible fluid flow, and equations describing reactive flow. Much of the existing theory for such nonlinear equations applies only to small solutions. However, large solutions are of great interest in applications such as gas flow, combustion, and detonations, and study of these solutions requires new techniques that will be developed in this project.