Regularity and Critical Thresholds Phenomena in Nonlinear Balance Laws

该项目的总体目标是研究与不同非线性平衡律相关的临界规律。我们专注于各种应用中感兴趣的边界情况，其中解决方案的内在特征，如光滑性与一般有限时间击穿，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。通常这种平衡是由流的全局不变量来维持的。这些包括频谱不变量，它反过来导致临界阈值现象，或边界不变量正则空间。这个项目的一个主要焦点是研究由欧拉动力学控制的平衡定律。这样的模型出现在不同的背景下，由不同的外力决定。先前的研究为研究这种欧拉动力学的临界阈值现象开发了一个精确的框架。在这里，我们寻求扩展，以包括由全球和非各向同性强迫驱动的更现实的模型。特别地，本项目研究了具有亚临界初始数据的多维Euler- poisson方程的全局正则性，以及具有初始构型的Euler和Navier-Stokes方程在边界正则空间中最小正则性的持久性。本研究计划旨在发展数学工具，以了解流体流动及其他动力过程的基本性质。这些结果将有潜在的应用于各种物理现象的建模和数值模拟，包括流体的湍流，这些现象是由所研究的偏微分方程类型控制的。

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.