Regularity and Critical Thresholds in Nonlinear Transport-Diffusion Equations

我们将使用现代数学工具，辅以新颖的计算模拟来研究正则化效应、临界阈值、时间衰减、熵稳定性和缩放现象。该项目将解决以下问题：(i)传输-扩散方程：扩散和熵耗散如何决定这些方程的正则化效果？（ii）欧拉动力学：旋转和压力之间的竞争如何决定整体稳定性？（iii）趋化性和相关的生物相关运输-扩散模型：我们应该如何解释超过临界时间的解决方案？（iv）图像的分层分解：如何将逆尺度空间与数据相适应，进行有效的图像处理？首席研究员和他的合作者计划继续开发新的分析和计算工具来探索运输-扩散模型，这有望有助于理解各种应用中现实模型的动力学。本项目的目标是研究非线性输运-扩散方程的全局特征的持久性，这在各种各样的应用中出现。例子包括具有退化扩散的非线性守恒定律，它模拟了沉降、交通流和图像处理中的数据驱动应用；无处不在的欧拉动力学控制着从半导体的小尺度到恒星形成的大尺度的一系列现象；以及生物应用中的趋化性模型。我们将注意力集中在基本传输-扩散方程的统一数学内容上。主要关注的是具有临界正则性的问题，这些问题取决于非线性对流机制、非线性扩散过程和可能驱动这种系统的非线性强迫之间的边界平衡。

We will use modern mathematical tools complemented by novel computational simulations to examine the phenomena of regularizing effects, critical thresholds, time decay, entropy stability, and scaling. The project will address the following questions: (i) Transport-diffusion equations: How do diffusion and entropy dissipation dictate the regularizing effect in such equations? (ii) Eulerian dynamics: How does the competition between rotation and pressure forcing determine the overall stability? (iii) Chemotaxis and related bio-related transport-diffusion models: How should we interpret the solutions beyond their critical time? (iv) Hierarchical decompositions of images: How can the inverse scale space be adapted to the data for effective image processing? The principal investigator and his collaborators plan to pursue the development of new analytical and computational tools to explore transport-diffusion models, which are expected to contribute to understanding of the dynamics of realistic models in a variety of applications. The goal of this project is to study the persistence of global features in nonlinear transport-diffusion equations, which arise in a wide variety of applications. Examples include nonlinear conservation laws with degenerate diffusion, which model sedimentation, traffic flows, and data-driven applications in image processing; the ubiquitous Eulerian dynamics governing a range of phenomena from the small scale of semi-conductors through the largest scale of star formation; and chemotaxis models found in biological applications. We focus our attention on the unifying mathematical content of the underlying transport-diffusion equations. Of primary interest are problems with critical regularity properties that hinge on a borderline balance between the nonlinear convection mechanisms, the nonlinear diffusion processes, and the possibly nonlinear forcing driving such a system.