Mathematical Questions in Kinetic Theory

这个研究项目将解决有关偏微分方程的解决方案，用于模拟物理学中的等离子体和流体的动力学的基本问题。重点将是研究物理学的基本定律，以预测大时间内宏观量的混合和弛豫。该项目将为偏微分方程理论和数学物理，动力系统和应用数学领域提供新的数学技术。此外，它将促进我们对等离子体物理和流体动力学中湍流的理解。该项目为研究生和其他早期职业研究人员提供培训机会。该项目旨在超越等离子体物理学和流体动力学中的混合和弛豫或朗道阻尼的研究，并解决有关有限规则性溶液行为的基本稳定性问题。所研究的基本方程包括等离子体物理中的经典Vlasov模型和流体动力学中的Euler和Navier-Stokes方程。其目的是提供新的见解朗道阻尼和混合等离子体和流体。主要的方法将涉及数学技术从频谱理论，预解式分析，傅立叶分析，色散偏微分方程，概率和统计物理。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

This research project will address fundamental problems concerning solutions to partial differential equations that are used to model the dynamics of plasmas and fluids in physics. The focus will be on studying the fundamental law of physics to predict mixing and relaxation of macroscopic quantities in the large time. The project will contribute new mathematical techniques to the theory of partial differential equations and the field of mathematical physics, dynamical systems, and applied mathematics. In addition, it will advance our understanding of turbulence in plasma physics and fluid dynamics. The project provides training opportunities for graduate students and other early-career researchers. The project seeks to advance beyond the study of mixing and relaxation, or Landau damping, in plasma physics and fluid dynamics, and to address fundamental stability problems concerning the behavior of solutions with limited regularity. The fundamental equations to be studied include the classical Vlasov models in plasma physics and the Euler and Navier-Stokes equations in fluid dynamics. The goal is to provide new insights into Landau damping and mixing in plasmas and fluids. The main approaches will involve mathematical techniques from spectral theory, resolvent analysis, Fourier analysis, dispersive PDEs, probability, and statistical physics.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.