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。该项目在色散偏微分方程领域进行研究。其意图是双重的。一方面，它关注某些色散方程解的行为的具体问题。另一方面，它专注于开发新的理论工具，自然地解决了广泛的问题，在该地区的哈密顿偏微分方程。这两个目标通过多线性谐波分析技术相互馈入。在该地区的主要问题是与解决方案，往往表现不同，这取决于他们的规律性是否是亚临界，临界或超临界相对于对称性的方程。特别是，希望该项目将补充和扩大我们对质量临界，质量亚临界和能量超临界方程的性质的理解。这个项目密切关注这些解决方案的定性行为。 这种行为基于解的局部/全局存在性、唯一性、正则性、光滑效应、有限时间爆破和解的渐近行为等性质。色散方程模拟自然界中发生的某些波动现象。他们的解往往是在无界域上空间上展开的波。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。一个著名的例子是非线性薛定谔方程。基于以前的进展，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测的情况下，以解释我们期望和观察到的数值特性。其目的是为这些物理观测提供分析测试依据。

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.