Nonlinear Transport, Degenerate Diffusion, Critical Regularity and Self-Organized Dynamics

该项目致力于以下五个方面的非线性时间相关问题。(i)欧拉动力学中的临界规律性：我们将使用谱动力学来研究无压欧拉方程的消失粘度解的新框架，受亚临界初始数据约束的欧拉-泊松方程的全局规律性，以及由无旋力驱动的浅水的长时间规律性。（ii）熵稳定性和良好平衡的浅水格式：我们将开发、分析和实现一类新的浅水方程的良好平衡格式。（iii）自组织动力学：我们将研究由速度对准驱动的模型的长期行为，并解决两个相互关联的问题。根据底层图的连通性，何时发生局部相互作用的群集，以及群集的水动力模型是如何实现的？我们还将探索自组织动力学的新模型，其中粒子间通信由它们的相对距离缩放。（iv）准线性输运-扩散方程中的正则化效应：我们将继续我们正在进行的在非线性标量守恒定律的具体设置和某些允许熵动力学公式的系统中使用速度平均的正则化效应的研究。(v)图像多尺度分解的积分微分方程：我们将研究用于图像去噪和去模糊的新型多尺度积分微分方程的局部化特性。本项目的最终目标是构建、分析和模拟由非线性偏微分方程（PDEs）控制的时变问题，并开发相关的新型计算方案。潜在的方程涉及非线性输运模型，自组织动力学，以及可能不同的小尺度分解为粒子动力学，动力学分布或像素强度；它们出现在各种应用中，包括流体动力学、集体行为科学、图像处理和去噪。我们将集中于统一方程的数学内容，使用现代分析工具和新的计算算法的协同作用，研究这些方程中全局特征的持久性。该项目通过研究为参与的研究生和博士后提供了良好的教育体验。

The project is devoted to the following five aspects of nonlinear time-dependent problems. (i) Critical regularity in Eulerian dynamics: we will use spectral dynamics to investigate a new framework for vanishing viscosity solutions of the pressure-less Euler equations, for global regularity of the Euler-Poisson equations subject to sub-critical initial data, and the long-time regularity of the shallow-water driven by irrotational forcing. (ii) Entropy stability and well-balanced shallow-water schemes: we will develop, analyze and implement a new class of well-balanced schemes for the shallow-water equations. (iii) Self-organized dynamics: we will study the long-time behavior of models driven by velocity-alignment and address two interrelated issues. When does flocking occur with local interactions, depending on the connectivity of the underlying graph, and how is it realized in hydrodynamic models of flocking? We will also explore new models of self-organized dynamics in which inter-particle communication is scaled by their relative distance. (iv) Regularizing effects in quasi-linear transport-diffusion equations: we will continue our ongoing research on regularizing effects using velocity averaging in the concrete setups of nonlinear scalar conservation laws and certain systems which admit an entropic kinetic formulation. (v) Integro-differential equations for multi-scale decomposition of images: we will study the localization properties of new multi-scale integro-differential equations for image de-noising and de-blurring. The ultimate goal of this project is to construct, analyze and simulate time-dependent problems which are governed by nonlinear Partial Differential Equations (PDEs) and develop related novel computational schemes. The underlying equations involve nonlinear transport models, self-organized dynamics, and possibly different small scale decompositions into particle dynamics, kinetic distributions, or intensity of pixels; they arise in diverse applications, including fluid dynamics, collective behavioral sciences, and image processing and de-noising. We will focus on the unifying mathematical content of the equations, using a synergy of modern analytical tools and novel computational algorithms, to study the persistence of global features in these equations. The project provides a great educational experience through research for the graduate students and postdoctoral fellows involved.