Volodymyr B. Taranov
Characteristic examples of continuous symmetries in hydrodynamic plasma theory (partial differential equations) and in kinetic Vlasov-Maxwell models (integro-differential equations) are considered. Possible symmetry extensions conditional and extended symmetries are discussed. Physical reasons for these symmetry extensions are clarified.
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The self-organized hydrodynamic models can be derived from the kinetic version of the Vicsek model. The formal derivations and local well-posedness of the macroscopic equations are done by Degond and his collaborators. In this paper, we rigorously justify this hydrodynamic limit.
We derive generalised multi-flow hydrodynamic reductions of the nonlocal kinetic equation for a soliton gas and investigate their structure. These reductions not only provide further insight into the properties of the new kinetic equation but also could prove to be representatives of a novel class of integrable systems of hydrodynamic type, beyond the conventional semi-Hamiltonian framework.
We prove an explicit, non-local hydrodynamic closure for the linear one-dimensional kinetic equation independent on the size of the relaxation time. We compare this dynamical equation to the local approximations obtained from the Chapman--Enskog expansion for small relaxation times. Our results rely on the spectral theory of Jacobi operators with rank-one perturbations.
In this paper we study the hydrodynamic limit of a B.G.K. like kinetic model on domains with boundaries via $BV_{loc}$ theory. We obtain as a consequence existence results for scalar multidimensional conservation laws with kinetic boundary conditions. We require that the initial and boundary data satisfy the optimal assumptions that they all belong to $L^1\cap L^\infty$ with the additional regularity assumptions that the initial data are in $BV_{loc}$. We also extend our hydrodynamic analysis to the case of a generalized kinetic model to account for forces effects and we obtain as a consequenc…
Given an obstacle in $\mathbb{R}^3$ and a non-zero velocity with small amplitude at the infinity, we construct the unique steady Boltzmann solution flowing around such an obstacle with the prescribed velocity as $|x|\to \infty$, which approaches the corresponding Navier-Stokes steady flow, as the mean-free path goes to zero. Furthermore, we establish the error estimate between the Boltzmann solution and its Navier-Stokes approximation. Our method consists of new $L^6$ and $L^3$ estimates in the unbounded exterior domain, as well as an iterative scheme preserving the positivity of the distribut…
The Kinetic and Hydrodynamic Bohm Criterions for Plasma Sheath Formation
该研究从数学角度探究等离子体鞘层形成的Bohm判据，证明Vlasov-Poisson系统边值问题可解性，明确动理学判据为可解必要条件并给出更简推导，同时通过δ质量极限研究揭示动理学与流体力学判据及两类系统解的关联。
The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman--Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad's moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particu…
This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, w…
Effective field theory of quasi-hydrodynamics from kinetic theory
本文从动理学理论出发构建线性准流体动力学的有效场论框架，证明守恒量与准守恒量可观测量的动力学可按快弛豫时间展开，零阶对应类Israel-Stewart的瞬态流体动力学普适描述，高阶修正满足微观理论的对称性等约束。
We study the evolution of a two component fluid consisting of ``blue'' and ``red'' particles which interact via strong short range (hard core) and weak long range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable choices of space-time scalings and system parameters we first obtain (formally) a mesoscopic kinetic Vlasov-Boltzmann equation for the one particle position and velocity distribution functions, appropriate for a description of the phase segregation kinetics in this system. Further scalings then yiel…
Gapless non-hydrodynamic modes in relativistic kinetic theory
本文提供了确定相对论动理论中非流体力学部分是否无隙的严格判据，证明当散射截面在高能下衰减为零时该部分无隙，揭示了相对论系统中普遍存在非热长寿命波。
Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2x2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a gener…
Hydrodynamic traffic flow models including random accidents: A kinetic derivation
本文从随机粒子模型形式化推导出含不连续通量的二阶双曲 PDE 宏观交通模型，并通过数值实验与粒子方法比较，利用解的统计量分析随机交通事故。
Learning the Optimal Hydrodynamic Closure
结合慢谱闭合理论与机器学习，从密度涨落数据学习广义输运系数，提出稀薄气体流动的最优流体动力学模型，显著优于先前本构定律，且不依赖克努森数小量假设。
Hydrodynamic models of preference formation in multi-agent societies
本文研究多智能体社会中偏好形成的流体动力学模型，通过从底层动力学模型推导局部平衡分布，将包含个人偏好微观变量的意见形成动力学方程过渡到流体动力学方程，数值实验揭示了意见与偏好区分在选择过程中的关键作用。
Particle, kinetic and hydrodynamic models for sea ice floes. Part II: Rotating floes with environmental forces
将非旋转浮冰的多尺度建模框架推广至旋转浮冰，引入非线性接触、科里奥利力和海洋倾斜，得到扩展的Vlasov型动理学方程及包含角动量平衡的流体动力学系统。
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