Nonlinear Dispersive PDE

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。该项目进行色散偏微分方程领域的研究。它的意图是双重的。一方面，它涉及关于某些色散方程的解的行为的具体问题。另一方面，它致力于发展新的理论工具，自然地解决该领域的哈密尔顿偏微分方程组的广泛问题。这两个目标通过多线性调和分析技术相互作用。该领域的主要问题与解有关，这些解往往表现不同，这取决于它们的正则性相对于方程的对称性是亚临界、临界还是超临界。特别是，希望该项目将补充和拓宽我们对质量临界方程、质量亚临界方程和能量超临界方程的性质的理解。本项目密切关注此类解决方案的定性行为。这种行为基于解的局部/整体存在性、唯一性、正则性、光滑化效应、有限时间爆破以及解的渐近行为等性质。他们的解决方案往往是在无界区域上传播的波。它们受到了数学家的极大关注，特别是因为它们在非线性光学、水波理论和等离子体物理中的应用。这一类的一个著名例子是非线性薛定谔方程。在以前进展的基础上，该项目的目标是将已知结果扩展到分析更困难、地形未知、现有理论结果落后于猜想的情况，以解释我们预期的和已经在数值上观察到的性质。其目的是为这些物理观测数据提供分析测试依据。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project conducts research in the area of dispersive partial differential equations. Its intent is twofold. On the one hand, it is concerned with specific questions about the behavior of solutions to certain dispersive equations. On the other, it is focused on developing new theoretical tools that naturally address a wide range of problems in the area Hamiltonian partial differential equations. These two objectives feed into each other by way of multilinear harmonic analysis techniques. The main problems in the area are associated with solutions that tend to behave differently depending on whether their regularity is subcritical, critical, or supercritical with respect to a symmetry of the equation. In particular, the hope is that the project will supplement and broaden our understanding about the properties of mass critical, mass subcritical, and energy supercritical equations. This project looks closely at the qualitative behavior of such solutions. This behavior is based on properties that include local/global existence of solutions, uniqueness, regularity, smoothing effect, finite-in-time blow up, and asymptotic behavior of solutions.Dispersive equations model certain wave phenomena that occur in nature. Their solutions tend to be waves that spread out spatially on unbounded domains. They have received a great deal of attention from mathematicians, in particular because of their applications to nonlinear optics, water wave theory, and plasma physics. One famous example from this class is the nonlinear Schrodinger equation. Building on the previous advances, the goal of the project is to extend the known results to cases where the analysis is harder, the terrain is unknown, and existing theoretical results lag behind conjecture for explaining the properties that we expect and have observed numerically. The intent is to provide analytical testing grounds for these physical observables.