Large Solutions to Systems of Nonlinear Equations

数学科学：非线性方程组的大解本研究的目的有三个方面。 首先，我们寻求一个更好的理解双曲（无粘）守恒律系统的解决方案，无论是大幅度或变化。 爆炸行为的新例子将被考虑，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，并建立具有球对称或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。 这是由描述化学过程的方程增强的Navier-Stokes方程建模的。 在这种情况下，我们特别感兴趣的稳定性的波动patterns.The工作涉及数学分析的解决方案，非线性偏微分方程。 研究将调查系统的守恒定律，可压缩流体流动，和方程描述反应流。 许多现有的理论，这种非线性方程只适用于小的解决方案。 然而，大型解决方案在气体流动，燃烧和爆炸等应用中具有很大的兴趣，这些解决方案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.