Critical Dispersive Partial Differential Equations

这个项目的主要目标是提高对色散偏微分方程的理解。色散偏微分方程包括波动方程、薛定谔方程和Korteweg de-Vries方程。这些方程在物理学中无处不在，从亚原子粒子的行为到星际引力波的现象都是如此。特别感兴趣的是解决方案的长期行为的问题。换句话说，给定一定的初始数据，方程是否存在解？如果存在一个解决方案，那么它会一直存在吗？当时间接近无穷大或解存在的最大时间时，解的行为是什么？我们能否对各种长时间行为进行编目，并对可能的现象进行完整的描述？该项目为研究生提供研究培训机会。在这个项目中，主要研究者和他的合作者研究了具有临界范数的初始数据的色散偏微分方程的长时间行为。许多不同的色散偏微分方程具有标度对称性，并且方程的解给出了整个解族。通常，标度对称性完全描述了方程的局部行为：方程对于临界空间中的初始数据是适定的，但对于不太规则（亚临界）空间中的数据是不适定的。我们希望了解这类方程在临界正则性下的长时间行为，已知在临界正则性下会发生局部适定性。此外，在许多这样的方程中，在次临界空间中出现不适定性的集合通常是测度为零的集合。因此，我们希望描述的性质的初始数据集的不适定性发生。在这个项目中解决的具体问题是薛定谔映射问题，聚焦，质量临界非线性薛定谔方程，能量亚临界非线性波和薛定谔方程，和一维立方非线性薛定谔方程。对于散焦，能量亚临界问题，我们希望散射发生在临界Sobolev空间的初始数据。孤立子是已知的薛定谔映射问题和质量临界非线性薛定谔方程。在这两种情况下，散射是已知的发生在孤子以下的初始数据（薛定谔映射，这只是在等变的情况下）。我们希望了解这些问题的解决方案的初始数据略高于孤立子。最后，对于一维非线性薛定谔方程，其目的是了解初始数据的长期行为，而不是在无穷远衰减。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The main objective of this project is to improve the understanding of dispersive partial differential equations. Dispersive partial differential equations include the wave, Schrodinger, and Korteweg de-Vries equation. These equations are ubiquitous in physics, modeling phenomena ranging from the behavior of subatomic particles to interstellar gravity waves. Of particular interest are questions of long time behavior of solutions. In other words, given certain initial data, does a solution to the equation exist? If a solution does exist, does it exist for all time? What is the behavior of the solution as time approaches either infinity or the maximum time for which the solution exists? Can we catalogue the various long time behaviors and obtain a complete description of the possible phenomena? The project provides research training opportunities for graduate students. In this project, the Principal Investigator and his collaborators study the long time behavior of dispersive partial differential equations with initial data in a critical norm. Many diverse dispersive partial differential equations have a scaling symmetry, and a solution to the equation gives an entire family of solutions. Often, the scaling symmetry completely describes the local behavior of the equation completely: the equation is well-posed for initial data in the critical space, but it is ill-posed for data in a less regular (subcritical) space. We wish to understand the long time behavior for such equations at the critical regularity, where it is known that local well-posedness occurs. Additionally, in many such equations, the set where ill-posedness occurs in a subcritical space is often a set of measure zero. Thus, we hope to describe the nature of the set of initial data for which ill-posedness occurs. The specific problems that are addressed in this project are the Schrodinger maps problem, the focusing, mass-critical nonlinear Schrodinger equation, the energy subcritical nonlinear wave and Schrodinger equations, and the one dimensional cubic nonlinear Schrodinger equation. For the defocusing, energy subcritical problems, we expect scattering to occur for initial data in the critical Sobolev space. Solitons are known to occur for the Schrodinger map problem and the mass-critical nonlinear Schrodinger equation. In both cases, scattering is known to occur for initial data below the soliton (for Schrodinger maps this is only in the equivariant case). We wish to understand the solution for initial data slightly above the soliton for these problems. Finally, for the one dimensional nonlinear Schrodinger equation, the aim is to understand the long time behavior for initial data that does not decay at infinity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.