Diffusion in Kinetic Equations

动力学方程模拟了相互作用的大粒子系统的密度演化。例如，它们可以用来研究气体或等离子体的演化。首席研究员(PI)感兴趣的是研究玻尔兹曼方程和朗道方程，这些方程是通过幂函数势相互排斥的粒子系统。这些方程表现出正则化效应。一个突出的悬而未决的问题是理解方程的自然流动是否会出现奇点，或者正则化效应是否实际上主导了演化并保持解的平稳。PI指导研究生和博士后对这个项目主题的研究。这个项目旨在开发非局部方程、抛物线方程和亚椭圆理论的分析工具，目标是它们在动力学方程中的应用。玻尔兹曼碰撞算符充当分数阶的非线性扩散算符。它可以在抛物型积分-微分方程组的框架下研究。朗道方程是统计力学中用于描述等离子体动力学的模型。它可以作为玻耳兹曼方程的一个极限情况，当掠过碰撞盛行时得到。它是一个二阶、非线性、抛物型方程。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-微分方程式的最新进展与统计力学中的经典玻尔兹曼方程联系起来。动力学方程包括一个关于速度的非线性扩散算子，以及一个关于空间的输运方程。所有变量的正则化效应都需要亚椭圆理论的思想。对于极软势情况下的Boltzmann方程，以及具有库仑势的Landau方程，方程的扩散部分在理论上不足以阻止解的爆破。在这种情况下，需要新的想法来正确理解方程的正规化影响。这一奖项反映了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 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.