Bounds and Asymptotic Dynamics for Nonlinear Evolution Equations

非线性演化方程的界和渐近动力学

基本信息

批准号：
2012333
负责人：
Andrei Tarfulea
金额：
$ 7.3万
依托单位：
Louisiana State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012333&HistoricalAwards=false
关键词：
Bounds Asymptotic Dynamics Nonlinear Evolution

项目摘要

Many physical, engineering, and biological phenomena are described by mathematical models with a large number of strongly-interacting particles. The range of these phenomena includes such diverse examples as complex and compressible fluids (combustion, aerospace engineering, and meteorology), nonlocal reaction-diffusion processes (nuclear physics, population biology, and genetics), and kinetic theory (plasma physics, swarm dynamics, and astrophysics). This project focuses on novel approaches to determining two fundamental characteristics of solutions to equations modeling large numbers of strongly interacting particles: their regularity and asymptotic behavior. The regularity of such problems establishes that the models are well-behaved, which often means the equations remain numerically tractable in computer simulations. The asymptotic theory seeks to find simplified limiting behavior for equations, in which many complex interactions average out and have a residual effect that governs the behavior of the system. Information about the limiting behavior is instrumental for applications such as medical imaging or materials science. For many important phenomena that demonstrate complex, nonlinear behavior, the application of known methods for analysis and control is greatly limited and not always possible. The aim of this project is to investigate three new techniques that partly overcome the difficulties caused by nonlinearity. The project will also provide training and research opportunities for both graduate and undergraduate students. The principal investigator will use techniques of nonlinear analysis, viscosity theory, and probability to establish bounds and asymptotic dynamics for the three major parts of the project. The first part focuses on exploring thermally enhanced dissipation for hydrodynamic equations where the viscosity grows with local temperature. From kinetic considerations and empirical observations, the kinematic viscosity of a compressible fluid flow increases with the local temperature and the local temperature is produced by friction. The intuition is that, in such models, regions of high turbulence self-regularize by producing hot spots which boost the viscosity exactly where it is needed to prevent the development of singularities. Prior work has identified this effect in two model problems (along with corresponding bounds). One of the main goals of the project is to push these types of estimates to physical models of compressible thermal fluids such as the Navier-Stokes-Fourier system, the equations of magneto-hydrodynamics, and the Poisson-Nernst-Planck-Fourier system for electrokinetic complex fluids. Enhanced thermal dissipation is a truly novel source of regularization compared to other known energy-based methods and lends itself naturally to dynamic weighted Sobolev estimates and entropy methods. The second part focuses on developing methods to extract asymptotic behavior from strongly nonlocal heterogeneous reaction-diffusion equations. There is a growing interest in extracting simpler macroscopic dynamics (often taking the form of geometric equations) from certain scaling limits of more complicated models. The nonlocal operators in these models present unique challenges in determining their residual impact on the (sometimes discontinuous) homogenized equation. The investigator plans to implement the techniques of viscosity theory to pursue homogenization phenomena for nonlocal periodic Fisher-KPP and bistable (Allen-Cahn) equations. The third part focuses on the regularity theory for kinetic equations (i.e., Landau and Boltzmann). Most regularity results for these equations rely on the assumption of having the lower bound on the density (as this often yields a minimum dissipation in the velocity variables). The investigator will explore the emergence of such lower bounds through probabilistic techniques, writing the kinetic equation as an approximate Fokker-Planck equation for a certain stochastic process.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

许多物理，工程和生物学现象是通过数学模型来描述的，这些模型具有大量的强烈相互交互的粒子。这些现象的范围包括复杂和可压缩流体（燃烧，航空工程和气象学），非局部反应 - 反应扩散过程（核物理学，种群生物学和遗传学）以及动力学理论（PLASMA物理学，SWARM动力学和天文学）等各种例子。该项目着重于确定解决方程的两个基本特征的新方法，以建模大量强烈相互作用的粒子：它们的规律性和渐近行为。此类问题的规律性确定了模型的行为良好，这通常意味着这些方程在计算机模拟中仍然可以在数值上拖延。渐近理论试图找到方程式的简化限制行为，其中许多复杂的相互作用平均消失，并具有控制系统行为的残留效应。有关限制行为的信息对于诸如医学成像或材料科学之类的应用至关重要。对于表现出复杂，非线性行为的许多重要现象，已知方法用于分析和控制的应用非常有限，并且并非总是可能的。该项目的目的是研究三种新技术，部分克服了非线性造成的困难。该项目还将为研究生和本科生提供培训和研究机会。主要研究者将使用非线性分析，粘度理论和可能性的技术来为项目的三个主要部分建立界限和渐近动力学。第一部分的重点是探索粘度随局部温度生长的流体动力方程的热增强耗散。从动力学的考虑和经验观察中，可压缩流体流动流的运动粘度随局部温度和局部温度增加而增加。直觉是，在这样的模型中，通过产生热点来提高粘度的高度湍流区域，从而促进了粘度，从而促进了防止奇异性发展所需的粘度。先前的工作已经在两个模型问题（以及相应的界限）中确定了这种效果。该项目的主要目标之一是将这些类型的估计值推到可压缩热流体的物理模型上，例如Navier-Stokes-tokes-tourier System，Magneto-Hydrodynalnicals的方程以及电动复合液的Poisson-Nernst-Nernst-Nernst-Nernst-Nernst-planck-Foursy System。与其他已知的基于能量的方法相比，增强的热耗散是一种真正新颖的正则化来源，并且自然而然地适合动态加权Sobolev估计和熵方法。第二部分的重点是开发从强烈非局部异质反应扩散方程中提取渐近行为的方法。从更复杂的模型的某些缩放限制中提取更简单的宏观动力学（通常采用几何方程式的形式）越来越感兴趣。这些模型中的非局部运算符在确定其对（有时不连续的）均质方程（有时是不连续的）方程的残余影响方面提出了独特的挑战。研究人员计划实施粘度理论的技术，以寻求非本地周期性的Fisher-KPP和Bistable（Allen-Cahn）方程的均质现象。第三部分侧重于动力学方程的规则性理论（即Landau和Boltzmann）。这些方程式的大多数规律性结果都取决于在密度上具有下限的假设（因为这通常会在速度变量中产生最小耗散）。研究人员将通过概率技术探索这种下限的出现，将动力学方程式写为一定随机过程的近似fokker-planck方程。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和更广泛影响的评估来评估的支持。