Well-Posedness and Long Time Behavior of Some Nonlinear Partial Differential Equations

拟开展的项目旨在解决数学物理领域基本非线性偏微分方程的重要问题，包括分别描述不可压缩流体流动和各种波动现象的Navier Stokes方程和非线性波动方程。在实践中，我们感兴趣的这些方程的解可能很简单，比如给定形状的船的阻力是多少，或者不同电磁波之间的非线性干扰变得太严重时。然而，这些简单问题的答案可能相当复杂，并且通常取决于对潜在数学方程的深刻理解。这些方程是非线性的，我们目前的理解从根本上来说还是不完整的。通常情况下，相对较少的“全局量”已经足以对复杂解的最有趣的方面提供满意的控制。在许多非线性偏微分方程的理论研究中，找到这些量和它们控制解的机制是一个最终目标。这些量可以在指导这些基本方程的实际应用中发挥重要作用，例如在设计计算解的数值方案时，使我们能够关注相对较少的重要参数，而忽略大量其他非必要参数。关于Navier Stokes的项目主要集中在以下几个问题上。1. 小初始旋流分量轴对称解的正则性。由于其他分量仍然很大，紧凑性参数不足以获得正则性，需要对解进行适当的动态控制。2. 与大尺度不变量解相关的谱假设。这种谱假设在研究非摄动区域的大尺度不变解时自然出现，并在Leray-Hopf弱解的唯一性问题中有深刻的应用。3. 稳态解的大距离渐近性。对于这样的问题，非线性是很重要的，即使是小的解。能量临界非线性波动方程的研究主要围绕各种模型的孤子分辨率猜想展开。对于聚焦能量临界波动方程和能量临界波图方程，PI的目标是在最近的部分结果的基础上，证明在非径向情况下沿时间序列的孤子分辨率猜想。PI还旨在在某些附加条件下建立完全的孤子分辨率，例如在一个气泡浓度的情况下，或在径向情况下。PI计划研究的另一个模型是在非径向情况下具有捕获势的散焦能量临界波动方程。在这种情况下，诸如“基态猜想”之类的基本问题仍然是开放的。更有趣的是，现在似乎可以严格地描述解在非摄动状态下的一般和非一般行为。PI计划解决其中的一些问题。

The proposed projects aim to solve important problems for fundamental nonlinear partial differential equations from areas of mathematical physics, including the Navier Stokes equations and the nonlinear wave equations, which describe the flow of incompressible fluid and a wide variety of wave phenomenon respectively. In practice, the questions about solutions to these equations that we are interested in can be simple, such as what is the drag force of a boat with a given shape, or when nonlinear interference between different electro-magnetic waves becomes too serious. The answers to these simple questions can however be quite complicated, and usually depend on deep understanding of the underlying mathematical equations. These equations are nonlinear, for which our current understanding is still fundamentally incomplete. As is often the case, relatively few ``global quantities" are already sufficient to provide satisfactory control on the most interesting aspects of the complicated solutions. It is an ultimate goal in many theoretic studies of nonlinear partial differential equations to find these quantities and the mechanism through which they control the solutions. These quantities can play an essential role in guiding practical applications of these fundamental equations, such as in the design of numerical schemes to calculate the solutions, by allowing us to focus on relatively few important parameters, while ignoring large amount of other non-essential parameters.The projects on Navier Stokes focus on the following problems. 1. The regularity of axi-symmetric solutions with small initial swirl component. Since the other components can still be large, compactness arguments are not sufficient to obtain regularity and suitable dynamical control on the solution is needed. 2. The spectral assumption related to large scale invariant solutions. Such spectral assumption appears naturally in the study of large scale invariant solutions in the non-perturbative regime, and has profound applications in the uniqueness problem of Leray-Hopf weak solutions. 3. Large distance asymptotics of steady state solutions. For such problems, it is well documented that nonlinearity is important even for small solutions. The projects on energy critical nonlinear wave equations are centered around the soliton resolution conjecture for various models. For the focusing energy critical wave equation and energy critical wave map equations, the PI aims to prove the soliton resolution conjecture along a sequence of times in the non-radial case, based on recent partial results. The PI also aims to establish full soliton resolution under certain additional conditions, such as in the case of one bubble concentration, or in the radial case. Another model the PI plans to study is the defocusing energy critical wave equation with a trapping potential, in the non-radial case. In this case, basic questions such as the ``ground state conjecture" are still open. More interestingly, it appears that one can now describe rigorously the generic and non-generic behavior of solutions in a non-perturbative regime. The PI plans to address some of these questions.