Global solutions of semilinear and quasilinear dispersive equations

PI将研究与某些演化方程光滑解的全局存在性和长期动力学相关的问题。更准确地说，PI将考虑几个拟线性模型，如欧拉-麦克斯韦系统，欧拉-泊松系统，以及2维和3维的无旋转水波问题。要考虑的主要问题与这些方程的某些平衡解的全局稳定性有关。在大数据的情况下，PI还将继续他在薛定谔图和其他自旋场系统的全球存在方面的工作。项目中考虑的方程描述了物理现象，如等离子体演化、铁磁模型和流体动力学，它们的相关性经常得到数值验证。PI建议严格研究这些方程的解，并将其行为的定量和定性信息恢复为数学定理。该提案的一个重要方面是支持研究生的培训，并促进与相关领域（如物理和工程）的研究人员的合作。

The PI will study questions related to global existence and long-term dynamics of smooth solutions of certain evolution equations. More precisely, the PI will consider several quasilinear models, such as the Euler-Maxwell system, the Euler-Poisson system, and the irrotational water wave problem in dimensions 2 and 3. The main problems to be considered have to do with the global stability of certain equilibrium solutions of these equations. The PI will also continue his work on the global existence of Schrodinger maps and other spin field systems, in the case of large data.The equations considered in the project describe physical phenomena, such as plasma evolutions, ferromagnetic models, and fluid dynamics, and their relevance is often verified numerically. The PI proposes to study the solutions of these equations rigorously, and recover quantitative and qualitative information about their behavior as mathematical theorems. An important aspect of this proposal is to support the training of graduate students and to foster collaborations with researchers in related fields, such as Physics and Engineering.