Diffusive Regularization in Kinetic and Fluid Equations

动力学方程构成了建模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，以及高速和低密度的流体流动，例如超音速流动。尽管这些模型很复杂，但它们经常可以看到热力学第二定律的表现，它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。这个项目探索了更精细的细节，以确定这些模型是否仍处于混沌状态，表现为湍流、冲击和等离子体回波，还是热化，变得更平滑并收敛到平衡状态。这些现象主要在两种情况下进行探讨：玻尔兹曼和朗道方程的正则性、延拓准则、潜在激波的形成，以及具有大规模电磁相互作用的版本；在有效粘度随局部湍流而增长的流体方程的增强扩散性中，由Kolmogorov提出并用于海洋学的一系列模型。该项目还为研究生、本科生和高中生提供培训和研究机会。本研究在两个重要的背景下探讨了新的规范化机制的构建和实施。首先，研究者将运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阶拉普拉斯算子，具有高度非局部和可能的简并系数。这些复杂性是适位性理论的主要障碍。然而，目前最先进的技术表明，只要某些宏观量保持在先验控制之下，光滑的唯一解就存在。研究人员最近的工作表明，这些数量中的一半实际上是动态控制的，从而对解决方案产生更精确的估计。该项目将这些结果扩展到更广泛的范围，具有边界的域，旋转对称构型和电磁相互作用的设置，并将它们与流体方程正则性理论的现有估计配对。其次，该项目将研究新的先验边界，该边界可以从局部温度影响粘度的非等温流体方程中导出。研究者之前的工作已经证明了一种独特的机制，即由热粘度引起的增强耗散，并为耦合的非等温模型开发了最大原理。这些影响在Navier-Stokes-Fourier系统和多孔介质类型和湍流耗散模型中进行了检验。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。

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.