Diffusive Regularization in Kinetic and Fluid Equations

动力学方程构成了模拟和理解具有大尺度相互作用的高能气体的数学基础，并用于预测等离子体的运动和辐射，例如，在工业和天文学中，以及在高速和低密度下的流体流动，例如，超音速流动。尽管这些模型很复杂，但它们经常看到热力学第二定律的表现，这将气体或流体推向最大熵的状态，这在统计学上更容易预测。该项目探索了更精细的细节，这些细节决定了这些模型是否保持在混沌状态，表现为湍流，冲击和等离子体回波，或者热化，变得更平滑并收敛到平衡状态。这些现象在两个主要的背景下进行了探讨：在规律性的属性，连续标准，潜在的冲击波形成的玻尔兹曼和朗道方程，和版本与大规模的电磁相互作用;和在增强扩散的流体方程，有效粘度增长与当地的湍流，一个家庭的模型起源于Kolmogorov和海洋学。该项目还为研究生、本科生和高中生提供了培训和研究机会。本研究在两个重要背景下考察了新型正则化机制的构建和实施。首先，研究人员将应用他们最近在动力学质量扩散方面的发现来探索玻尔兹曼和朗道方程的正则性程序的当前前沿。对于这些高能气体和等离子体模型，碰撞相互作用的行为大致类似于分数拉普拉斯算子，具有高度非局部性和可能的简并系数。这些复杂性是适定性理论的主要障碍。然而，目前的国家的最先进的赠款，顺利唯一的解决方案存在，只要某些宏观量保持在控制下的先验。研究人员最近的工作确定，这些数量中的一半实际上是动态控制的，从而为解决方案提供更精确的估计。该项目将这些结果扩展到更广泛的范围，具有边界的域，旋转对称配置和具有电磁相互作用的设置，并将它们与流体方程的正则性理论的现有估计配对。第二，该项目将研究新的先验边界，可以从非等温流体方程中推导出，其中局部温度影响粘度。研究人员先前的工作已经证明了一个独特的机制，增强耗散所产生的热粘度和发展最大原则耦合非等温模型。这些影响在纳维尔-斯托克斯-傅立叶系统和多孔介质类型和湍流耗散模型中进行了研究。该奖项反映了NSF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。

Kinetic equations form the mathematical basis for modeling and understanding high-energy gases with large-scale interactions and are used to predict the motion and radiation of plasmas, e.g., in industry and astronomy, as well as fluid flows at high speed and low density, for example, supersonic flows. Despite their complexity, these models often see manifestations of the second law of thermodynamics, which push the gas or fluid towards a state of maximum entropy, which is, statistically, easier to predict. This project explores the finer details that determine whether such models remain in a chaotic regime, manifesting as turbulence, shocks, and plasma echoes, or thermalize, becoming smoother and converging to an equilibrium. These phenomena are explored in two main contexts: in the regularity properties, continuation criteria, potential shock formation of the Boltzmann and Landau equations, and versions with large-scale electromagnetic interactions; and in the enhanced diffusivity of fluid equations where effective viscosity grows with local turbulence, a family of models originated by Kolmogorov and used in oceanography. The project also provides training and research opportunities for graduate, undergraduate, and high school students.This research examines the construction and implementation of novel regularizing mechanisms in two important contexts. First, the investigator will apply their recent discoveries in kinetic mass spreading to probe the current frontier of the regularity program for the Boltzmann and Landau equations. For these models of high-energy gases and plasmas, the collision interaction is known to behave roughly like a fractional Laplacian operator with highly nonlocal and possibly degenerate coefficients. These intricacies are major impediments to the well-posedness theory. Nevertheless, the current state-of-the-art grants that smooth unique solutions exist for as long as certain macroscopic quantities remain under control a priori. The investigator's recent work establishes that half of these quantities are in fact controlled dynamically, yielding more precise estimates for the solution. This project extends these results to wider scopes, domains with boundary, rotationally symmetric configurations, and settings with electromagnetic interactions, and pairs them with existing estimates from the regularity theory for fluid equations. Second, the project will investigate novel a priori bounds that can be derived from non-isothermal fluid equations where the local temperature influences the viscosity. The investigator's prior work has demonstrated a unique mechanism for enhanced dissipation arising from thermal viscosity and in developing maximum principles for coupled non-isothermal models. These effects are examined in the Navier-Stokes-Fourier system and in models of porous media type and of turbulent dissipation.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.