Diffusive Regularization in Kinetic and Fluid Equations

动力学方程构成了模拟和理解具有大规模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。尽管它们很复杂，但这些模型经常看到热力学第二定律的表现，热力学第二定律将气体或流体推向最大熵状态，从统计学上讲，这一状态更容易预测。这个项目探索了决定这些模型是否保持在混乱状态的更细微的细节，表现为湍流、冲击和等离子体回声，或者热化，变得更平滑并收敛到平衡。这些现象是在两个主要背景下探讨的：在Boltzmann方程和Landau方程的正则性、连续性准则、势激波形成以及具有大尺度电磁相互作用的版本中；在有效粘性随局部湍流增长的流体方程的增强扩散性中，由Kolmogorov创立的用于海洋学的一族模型。该项目还为研究生、本科生和高中生提供培训和研究机会。本研究在两个重要背景下考察了新的规范化机制的构建和实施。首先，研究人员将应用他们在动力学质量扩散方面的最新发现来探索玻尔兹曼方程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，已知碰撞相互作用的行为大致类似于分数拉普拉斯算子，具有高度的非局域系数和可能的简并系数。这些错综复杂的问题是适当性理论的主要障碍。尽管如此，目前最先进的解决方案是，只要某些宏观数量先验地处于控制之下，平滑的独特解决方案就会存在。这位研究人员最近的工作证明，这些量中的一半实际上是动态控制的，从而对解决方案产生了更准确的估计。该项目将这些结果扩展到更广泛的范围，具有边界、旋转对称构型的区域，以及具有电磁相互作用的环境，并将它们与流体方程的正则性理论的现有估计配对。其次，该项目将研究可以从局部温度影响粘度的非等温流体方程中推导出的新的先验界限。研究人员先前的工作已经证明了一种独特的机制，用于增强热粘性引起的耗散，并在开发耦合非等温模型的最大值原理方面。这些影响在纳维-斯托克斯-傅立叶系统以及多孔介质类型和湍流耗散的模型中进行了研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.