Bounds and Asymptotic Dynamics for Nonlinear Evolution Equations

许多物理、工程和生物现象都是由具有大量强相互作用粒子的数学模型来描述的。这些现象的范围包括复杂和可压缩流体（燃烧，航空航天工程和气象学），非局部反应扩散过程（核物理学，种群生物学和遗传学）和动力学理论（等离子体物理学，群动力学和天体物理学）等不同的例子。这个项目的重点是新的方法来确定两个基本特征的方程建模大量强相互作用粒子的解决方案：他们的规律性和渐近行为。这类问题的规律性决定了模型的行为良好，这通常意味着方程在计算机模拟中仍然是数值上易于处理的。渐近理论试图找到方程的简化极限行为，其中许多复杂的相互作用达到平均，并具有控制系统行为的剩余效应。关于极限行为的信息对于诸如医学成像或材料科学的应用是有用的。对于许多表现出复杂的非线性行为的重要现象，用于分析和控制的已知方法的应用受到很大限制，并且并不总是可能的。本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。主要研究人员将使用非线性分析，粘性理论和概率的技术，以建立该项目的三个主要部分的界限和渐近动力学。第一部分着重于探讨热增强耗散流体动力方程的粘度增长与当地温度。从动力学考虑和经验观察，可压缩流体流动的运动粘度随着局部温度而增加，并且局部温度由摩擦产生。直觉是，在这样的模型中，高湍流区域通过产生热点而自我调节，这些热点恰好在需要防止奇点发展的地方提高粘度。先前的工作已经在两个模型问题中确定了这种效应（沿着相应的边界）。该项目的主要目标之一是将这些类型的估计推到可压缩热流体的物理模型中，例如Navier-Stokes-Fourier系统，磁流体动力学方程和用于电动复杂流体的Poisson-Nernst-Planck-Fourier系统。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，并且自然地适用于动态加权Sobolev估计和熵方法。第二部分着重于发展从强非局部非均匀反应扩散方程中提取渐近行为的方法。人们对从更复杂模型的某些缩放限制中提取更简单的宏观动力学（通常采用几何方程的形式）的兴趣越来越大。这些模型中的非局部算子在确定其对（有时是不连续的）均匀化方程的剩余影响方面提出了独特的挑战。研究人员计划实施的粘性理论的技术，追求非局部周期性Fisher-KPP和Allen-Cahn方程的均匀化现象。第三部分着重于动力学方程的正则性理论（即，朗道和玻尔兹曼）。这些方程的大多数正则性结果依赖于密度有下限的假设（因为这通常会产生速度变量的最小耗散）。研究人员将通过概率技术探索这种下限的出现，将动力学方程写成近似的Fokker-Planck方程，用于某个随机过程。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.