Bounds and Asymptotic Dynamics for Nonlinear Evolution Equations

许多物理、工程和生物现象都是用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、群体动力学和天体物理学）等不同的例子。这个项目的重点是用新的方法来确定模拟大量强相互作用粒子的方程解的两个基本特征：它们的规律性和渐近行为。这类问题的规律性表明模型表现良好，这通常意味着在计算机模拟中方程在数值上仍然易于处理。渐近理论寻求找到方程的简化极限行为，其中许多复杂的相互作用平均并具有控制系统行为的残余效应。关于极限行为的信息对于医学成像或材料科学等应用是有用的。对于许多表现出复杂、非线性行为的重要现象，应用已知的分析和控制方法是非常有限的，而且并不总是可能的。该项目的目的是研究三种新技术，这些技术部分地克服了非线性引起的困难。该项目还将为研究生和本科生提供培训和研究机会。首席研究员将使用非线性分析、粘度理论和概率技术，为项目的三个主要部分建立边界和渐近动力学。第一部分着重探讨黏度随局部温度增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，可压缩流体流动的运动粘度随局部温度的增加而增加，而局部温度是由摩擦产生的。直观的感觉是，在这样的模型中，高湍流区域通过产生热点来实现自正则化，这些热点恰恰在需要防止奇点发展的地方提高了粘度。先前的工作已经在两个模型问题中确定了这种效应（以及相应的边界）。该项目的主要目标之一是将这些类型的估计推广到可压缩热流体的物理模型中，如Navier-Stokes-Fourier系统、磁流体动力学方程和电动复杂流体的泊松-能斯特-普朗克-傅里叶系统。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，它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相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极限中提取更简单的宏观动力学（通常采用几何方程的形式）越来越感兴趣。这些模型中的非局部算子在确定它们对（有时不连续的）均匀方程的残余影响方面提出了独特的挑战。研究者计划应用粘度理论技术来研究非局部周期Fisher-KPP和双稳态（Allen-Cahn）方程的均匀化现象。第三部分着重于动力学方程（即朗道和玻尔兹曼）的正则性理论。这些方程的大多数规律性结果依赖于密度的下界假设（因为这通常会产生速度变量的最小耗散）。研究者将通过概率技术探索这种下界的出现，将动力学方程写成某个随机过程的近似Fokker-Planck方程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.