Theory and Application of Kinetic Equations

我提出的研究计划集中在动力学方程的分析、应用和数值计算上。*在分析和计算部分，我打算在三个主要方面进行研究：*1.非线性Boltzmann方程的适定性：我们将与R.Alonso，Y.Morimoto和T.Yang一起研究具有非截断核的非线性Boltzmann方程的适定性。我们的主要目标是放松当前文献中对解的光滑性和快速衰减性的假设，并寻找能够保持唯一性的尽可能大的空间。我们还将研究从粗略数据开始的非截止Boltzmann方程的解的正则化。*2.非线性边界层动力学方程：边界层动力学方程在确定作为动力学方程流体动力学极限的流体方程(如Navier-Stokes和Euler)的适当边界条件时起着至关重要的作用。目前关于气体动力学中这些方程的大部分理论工作都局限于线性情况。与Li和J.Lu一起，我们旨在严格证明不同参数区域上的非线性边界层Boltzmann方程解的存在性。我们还将设计有效的数值方法来求解非线性方程，并将其与内部的流体方程耦合。*3.固体在气体中的运动：这项研究是研究运动的固体与周围气体的相互作用。我将研究外力作用于固体的环境，气体由各种动力学方程模拟。目前的理论工作大多局限于自由流动气体。我的目标是研究碰撞气体，严格描述固体的长期渐近行为，并与现有的数值结果进行比较。*同时，我建议致力于数学生物学中的动力学模型，重点放在两个主要应用上：*5.聚集行为：动力学和流体类型的方程已被广泛应用于模拟鸟类等代理人的聚集行为。与R.Fetecau和H.Huang一起，我们建议研究随机聚集模型的平均场极限。目前在这个方向上的结果要么覆盖整个空间，要么具有相对规则的相互作用势。我们的目标是扩展到在有边界的区域上设置奇异势的模型。*4.趋化性：趋化性研究细菌在外部环境影响下的运动。与B.Perthame和M.Tang一起，我们将研究非经典动力学传输方程，该方程模拟了具有内态和噪声的细菌密度的演化。我们的主要目标是了解*这些方程在不同区域所涉及的生物参数。我们将推导出不同区域的极限方程和可测量量，并将它们与实验数据进行比较。

My proposed research program are focused on analysis, application, and numerical computation of kinetic equations. ***In the analysis and computation part, I propose to conduct research in three main subjects:***1. Well-posedness of nonlinear Boltzmann equations: Together with R. Alonso, Y. Morimoto, and T. Yang, we will investigate the well-posedness of nonlinear Boltzmann equations with non-cutoff kernels. Our main goal is to relax the smoothness and fast-decay assumptions for solutions in the current literature and look for spaces that are as large as possible where uniqueness can hold. We will also investigate regularization of solutions for non-cutoff Boltzmann equations starting from rough data. ***2. Nonlinear boundary layer kinetic equation: Boundary layer kinetic equations play a crucial role in determining the proper boundary conditions for fluid equations (such as Navier-Stokes and Euler) derived as the hydrodynamic limits of kinetic equations. Most of the current theoretical work regarding these equations in gas dynamics are restricted to the linear case. Together with Q. Li and J. Lu, we aim at rigorously justifying the existence of the solutions for nonlinear boundary layer Boltzmann equations for various parameter regions. We will also design efficient numerical methods for solving nonlinear equations and couple them with fluid equations in the interior. ***3. Motion of a solid in a gas: This research is to study the interaction of a moving solid with surrounding gases. I will study the setting where there is an external force acting on the solid and the gases are modelled by various kinetic equations. Current theoretical works are mostly restricted to free streaming gases. My goal is to study collisional gases and rigorously characterize the long-time asymptotic behaviour of the solid and compare with existing numerical results. ***Simultaneously, I propose to work on kinetic models in math biology, focusing on two major applications:***5. Aggregation behaviour: Kinetic and fluid types of equations have been widely applied in modelling aggregation behaviour of agents like birds. Together with R. Fetecau and H. Huang, we propose to study mean field limits of stochastic aggregation models. Current results in this direction are either over the whole space or with relatively regular interaction potentials. Our goal is to extend to models with singular potentials posed on domains with boundaries.***4. Chemotaxis: Chemotaxis studies the motion of bacteria under the influence of exterior environment. Together with B. Perthame and M. Tang, we will study non-classical kinetic transport equations modelling the evolution of the density of bacteria with internal states and noise. Our main goal is to understand ***these equations in different regions of the biological parameters involved. We will derive limiting equations and measurable quantities in different regions and compare them with experimental data.