Fundamental aspects of inverse kinetic theories for incompressible Navier-Stokes equations concern the possibility of defining uniquely the kinetic equation underlying such models and furthermore, the construction of a kinetic theory implying also the energy equation. The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations. These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory. In this paper we intend to prove that the non-uniqueness feature can be resolved by imposing suitable assumptions. These include, in particular, the requirement that the kinetic equation be equivalent, in a suitable sense, to a Fokker-Planck kinetic equation. Its Fokker-Planck coefficients are proven to be uniquely determined by means of appropriate prescriptions. In addition, as a further result, it is proven that the inverse kinetic equation satisfies both an entropy principle and the energy equation for the fluid fields.
Fokker-Planck Kinetic description of small-scale fluid turbulence for classical incompressible fluids
扩展的统计方法提出了一个平行的文件\cite % {Tessarotto 2008-aa}，这项工作的目的是提出一个随机逆动力学理论的基础上引入一个合适的\textit{local phase-space probability density function}（pdf）的小尺度流体动力学湍流。特别是，我们提出的问题的建设福克-普朗克动力学模型的水动力湍流。这里采用的方法是基于所谓的IKT方法（逆动力学理论），由Ellero等人（2004-2008）开发的，其允许基于采用局部pdf的不可压缩流体的精确相空间描述。我们打算表明，对于规定的随机模型，本方法允许确定唯一的随机流体场的时间演化。随机平均的本地PDF服从动力学方程，虽然一般非马尔可夫，局部速度空间中可以近似通过一个合适的福克-普朗克动力学方程。作为副作用，相同的pdf被证明通常具有非高斯行为。
A discrete unified gas-kinetic scheme for immiscible two-phase flows
在这项工作中，我们扩展了离散统一气体动力学方案（DUGKS）[Guo et al.，E88，033305（2013）]以继续两相流。在DUGKS框架中，使用两个动力学模型方程来求解准不可压缩相场控制方程[Yang et al.，E 93，043303（2016）]。一个是Chan-Hilliard（CH）方程，另一个是纳维-斯托克斯方程。通过Chapman-Enskog分析，DUGKS方法可以正确地恢复准不可压缩相场控制方程。与以前的基于相场的LB模型不同，DUGKS中的Courant-Friedricks-Lewy条件是可调的，这可以增加数值稳定性。此外，有限体积公式可以很容易地实现非均匀网格，可以提高数值精度的模型。通过模拟静止液滴、分层Poiffille流、上升气泡和Rayleigh-Taylor不稳定性，并与准不可压格子玻尔兹曼方法（LBM）进行比较，验证了该模型的有效性。数值计算结果表明，该方法能够高精度、稳定地跟踪界面。该模型也能够处理更广泛的粘度和密度比的准不可压缩格子玻尔兹曼模型。该模型是一种很有前途的两相流数值模拟工具。
Model of incompressible turbulent flows via a kinetic theory
State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology（华中科技大学能源与动力工程学院煤燃烧国家重点实验室）；Institute of Interdisciplinary Research for Mathematics and Applied Science, Huazhong University of Science and Technology（华中科技大学数学与交叉科学研究院）；College of Energy Engineering, Zhejiang University（浙江大学能源工程学院）
