Diffusive Regularization in Kinetic and Fluid Equations

动力学和流体方程的扩散正则化

基本信息

批准号：
2108209
负责人：
Andrei Tarfulea
金额：
$ 19.87万
依托单位：
Louisiana State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108209&HistoricalAwards=false
关键词：
Diffusive Regularization Kinetic Fluid Equations

项目摘要

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.

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，以及以高速和低密度流动的流体流动，例如超音速流。尽管它们的复杂性，但这些模型经常会看到热力学第二定律的表现，这些定律将气体或流体推向最大熵的状态，从统计上讲，这很容易预测。该项目探讨了确定此类模型是否保持在混乱状态的细节，表现为湍流，冲击和等离子体的回声或热层，变得更顺畅并融合到平衡中。这些现象在两个主要情况下进行了探索：在规则性特性，持续标准，玻尔兹曼和兰道方程的潜在冲击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增长的流体方程增强的扩散率中，一个由Kolmogorov起源于海洋学的模型家族。该项目还为研究生，本科和高中生提供了培训和研究机会。本研究研究了在两种重要情况下进行新颖的正规机制的建设和实施。首先，研究人员将在动力学质量扩散中应用他们的最新发现，以探究玻尔兹曼和兰道方程规则性计划的当前前沿。对于这些高能气体和等离子体的模型，已知碰撞相互作用的行为就像是分数拉普拉斯运算符，具有高度非局部性和可能的退化系数。这些复杂性是适当性理论的主要障碍。然而，只要某些宏观数量仍处于先验的控制之下，就会存在平滑独特解决方案的当前最新赠款。研究者最近的工作表明，这些数量的一半实际上是动态控制的，对解决方案产生了更精确的估计。该项目将这些结果扩展到更广泛的范围，具有边界，旋转对称配置的域以及具有电磁相互作用的设置，并将它们与流体方程的规则性理论中的现有估计配对。其次，该项目将研究可从非等温流体方程得出的新型先验边界，其中局部温度会影响粘度。研究者的先前工作证明了一种独特的机制，可增强耗散粘度以及为耦合非等温模型的最大原理引起的耗散。这些效果在Navier-Stokes-tour的系统以及多孔媒体类型和动荡的耗散模型中进行了研究。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，认为值得通过评估来获得支持。