Regularity and Critical Thresholds Phenomena in Nonlinear Balance Laws

这个项目的总体目标是研究与不同的非线性平衡律相关的临界规律性。 我们专注于边界情况，在各种应用中感兴趣，其中解决方案的内在特征，如平滑度与通用有限时间崩溃，时间衰减等，取决于非线性对流和各种（可能是非线性的）强迫机制之间的微妙平衡。 这种平衡通常由流的全局不变量来维持。 这些包括光谱不变量，这反过来又导致临界阈值现象，或边界不变的正则空间。 该项目的一个主要焦点主题是欧拉动力学控制的平衡定律的研究。 这种模式出现在不同的背景下，由不同的强迫决定。 以前的研究开发了一个精确的框架，研究临界阈值现象，这样的欧拉动力学。 在这里，我们寻求扩展，包括更现实的全球和非各向同性强迫驱动的模型。 本研究课题主要研究亚临界初始状态下的多维Euler-Poisson方程的整体正则性，以及边界正则空间中初始状态下的Euler方程和Navier-Stokes方程的最小正则性的持续性。本研究课题旨在开发用于理解流体流动等动力学过程的基本性质的数学工具。 结果将有潜在的应用程序的建模和数值模拟的各种物理现象，包括流体的湍流，所研究的偏微分方程的类型。

The overall goal of this project is the study of critical regularity associated with different nonlinear balance laws. We focus on borderline cases, of interest in various applications, where intrinsic features of the solutions such as smoothness vs. generic finite-time breakdown, time decay, etc., hinge on a delicate balance between nonlinear convection and a variety of (possibly nonlinear) forcing mechanisms. Often this balance is maintained by global invariants of the flow. These include spectral invariants, which in turn lead to critical threshold phenomena, or borderline invariant regularity spaces. A main focal topic of this project is the study of balance laws governed by Eulerian dynamics. Such models show up in different contexts dictated by different forcing. Previous research developed a precise framework for studying the critical threshold phenomena for such Eulerian dynamics. Here we seek extensions to include more realistic models driven by global and non-isotropic forcing. In particular, this project investigates global regularity for multidimensional Euler-Poisson equations with sub-critical initial data and persistence of minimal regularity for Euler and Navier-Stokes equations with initial configurations in borderline regularity spaces.This research project pursues the development of mathematical tools for the understanding of fundamental properties of fluid flow and other dynamical processes. The results will have potential application for the modeling and numerical simulation of a variety of physical phenomena, including the turbulent flow of fluids, that are governed by the type of partial differential equations under study.