Regularity and Critical Thresholds in Nonlinear Transport-Diffusion Equations

我们将使用现代数学工具并辅之以新的计算模拟来研究正则化效应、临界阈值、时间衰减、熵稳定性和标度等现象。该项目将解决以下问题：(1)运输-扩散方程：扩散和熵耗散如何规定这种方程中的正则化效应？(2)欧拉动力学：旋转和压力之间的竞争如何决定总体稳定？(3)趋化性和与生物有关的运输-扩散模型：我们应该如何解释这些解决办法，使其超越其关键时期？(4)图像的分层分解：如何使逆尺度空间适应数据以进行有效的图像处理？首席研究员和他的合作者计划开发新的分析和计算工具，以探索传输-扩散模型，预计这将有助于理解各种应用中现实模型的动力学。这个项目的目标是研究在广泛应用中出现的非线性传输扩散方程的全局特征的持久性。例子包括退化扩散的非线性守恒定律，它模拟了图像处理中的沉积、交通流量和数据驱动的应用；无处不在的欧拉动力学控制着从小范围的半导体到最大规模的恒星形成的一系列现象；以及在生物应用中发现的趋化性模型。我们把注意力集中在基本输运-扩散方程的统一数学内容上。主要感兴趣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样一个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.