Diffusion in Kinetic Equations

动力学方程模拟了一个大的相互作用粒子系统的密度演化。例如，它们可用于研究气体或等离子体的演化。首席研究员（PI）对波尔兹曼方程和朗道方程的研究感兴趣，这些方程是由幂律势相互排斥的粒子系统组成的。这些方程表现出正则化效应。一个突出的开放问题是理解是否奇点可以从方程的自然流动中出现，或者是否正则化效应实际上主导了演化并保持解的平滑。项目负责人指导研究生和博士后进行本课题的研究。本项目旨在开发非局部方程、抛物方程和准椭圆理论的分析工具，并将其应用于动力学方程。玻尔兹曼碰撞算子是分数阶非线性扩散算子。它可以在抛物型积分微分方程的框架内进行研究。朗道方程是统计力学中用来描述等离子体动力学的一个模型。它可以作为玻尔兹曼方程在掠掠碰撞发生时的极限情况得到。它是一个二阶非线性抛物方程。该项目将数学和数学物理的不同领域联系起来，将非线性积分-微分方程的最新进展与统计力学中的经典玻尔兹曼方程联系起来。动力学方程包含一个关于速度的非线性扩散算子，结合一个关于空间的输运方程。所有变量的正则化效果需要准椭圆理论的思想。对于具有非常软势的玻尔兹曼方程，以及具有库仑势的朗道方程，方程的扩散部分在理论上不足以阻止解的爆炸。在这种情况下，需要新的想法来正确理解方程的正则化效果。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 model the evolution of densities of a large system of interactive particles. They may be used, for example, to study the evolution of a gas or a plasma. The Principal Investigator (PI) is interested in the study of the Boltzmann and Landau equations, for systems of particles that repel each other by power-law potentials. These equations exhibit a regularization effect. An outstanding open problem is to understand if a singularity could emerge from the natural flow of the equation, or if the regularization effects actually dominate the evolution and keep the solutions smooth. The PI mentors graduate students and postdocs in research on the topics of this project.This project aims at developing tools in the analysis of nonlocal equations, parabolic equations and hypoelliptic theory targeted to their applications in kinetic equations. The Boltzmann collision operator acts as a nonlinear diffusive operator of fractional order. It can be studied in the framework of parabolic integro-differential equations. The Landau equation is a model from statistical mechanics used to describe the dynamics of plasma. It can be obtained as a limit case of the Boltzmann equation when grazing collisions prevail. It is a second order, nonlinear, parabolic equation. The project connects different areas of mathematics and mathematical physics, relating recent progress in nonlinear integro-differential equations with the classical Boltzmann equation from statistical mechanics. Kinetic equations involve a nonlinear diffusive operator with respect to velocity, combined with a transport equation with respect to space. The regularization effect in all variables requires ideas from hypoelliptic theory. For the Boltzmann equation in the case of very soft potentials, as well as for the Landau equation with Coulomb potentials, the diffusive part of the equations is not strong enough to prevent the solution from blowing up in theory. In that case, new ideas are needed to properly understand the regularization effects of the equation.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.