Sa Jun Park, Seok-Bae Yun
We establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level.
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In this paper, we show that the entropy production functional for the polyatomic ellipsoidal BGK model can be decomposed into two non-negative parts. Two applications of this property: the $H$-theorem for the polyatomic BGK model and the weak compactness of the polyatomic ellipsoidal relaxation operator, are discussed.
Recently, a novel relativistic polyatomic BGK model was suggested by Pennisi and Ruggeri [J. of Phys. Conf. Series, 1035, (2018)] to overcome drawbacks of the Anderson-Witting model and Marle model.In this paper, we prove the unique existence and asymptotic behavior of classical solutions to the relativistic polyatomic BGK model when the initial data is sufficiently close to a global equilibrium.
We study the entropy production estimate for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the time evolution of polyatomic particle systems. An interesting dichotomy is observed between $0<θ\leq 1$ and $θ=0$: In each case, a distinct target Maxwellians should be chosen to estimate the entropy production functional from below by the relative entropy. The time asymptotic equilibrium state toward which the distribution function stabilizes bifurcates accordingly.
We consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal BGK model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic Maxwellian. We observe that the linearized relaxation operator is decomposed into a trully polyatomic part and a essentially monatomic part, leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter $θ$ reaches zero. Accordingly, we employ two different sets of micro-macro system to derive the full c…
Relativistic BGK model of Marle for polyatomic gases near equilibrium
本文将多原子气体相对论扩展热力学理论应用于Marle提出的相对论BGK模型，通过在广义Jüttner分布附近进行扰动分析并研究线性算子性质，证明了初值充分接近全局平衡态时经典解的全局存在性及长时间行为。
We consider the socalled Bathnagar-Gross-Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of mixture a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom li…
We consider two models for a two component gas mixture with translational and internal energy degrees of freedom described by a BGK approximation assuming that the number of particles of each species remains constant. The two species are allowed to have different degrees of freedom in internal energy and are modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the species. We consider two models in the literature and prove that the models satisfy a different equation of state than the ideal gas law. Moreover, we prove…
Kinetic models for polyatomic gases have two temperatures for the two different types of degrees of freedom, the translational and the internal energy degrees of freedom. Therefore, in the case of BGK models one expects two types of relaxations, a relaxation of the distribution function to a Maxwell distribution and a relaxation of the two temperatures to an equal value. The speed for the first type of relaxation may be faster or slower than the second type of relaxation. Models found in the literature often allow only for one of these two cases. We believe that a model should allow for both c…
The ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model of the Boltzmann equation designed to yield the correct Prandtl number in the Navier-Stokes limit. In this paper, we make two observations on the entropy production functional of the ES-BGK model. First, we show that the Cercignani type estimate holds for the ES-BGK model in the whole range of parameter $-1/2<ν<1$. Secondly, we observe that the ellipsoidal relaxation operator satisfies an unexpected sign-definite property. Some implications of these observations are also discussed.
We address the boundary value problem for the ellipsoidal BGK model of the Boltzmann equation posed in a bounded interval. The existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity, and the gas is sufficiently rarefied.
The ellipsoidal BGK model was introduced in \cite{Ho} to fit the correct Prandtl number in the Navier-Stokes approximation of the classical BGK model. In the paper we establish the global existence of mild solutions to the Cauchy problem on the model for a class of initial data allowed to have large oscillations. The proof is motivated by a recent study of the same topic on the Boltzmann equation in \cite{DHWY}.
Determination of equilibrium parameters of the Marle model for polyatomic gases
本文为多原子气体引入基于Jüttner分布的相对论Marle模型，并证明其平衡参数的唯一存在性。
We propose an extension of the Ellipsoidal-Statistical BGK model to account for discrete levels of vibrational energy in a rarefied polyatomic gas. This model satisfies an H-theorem and contains parameters that allow to fit almost arbitrary values for the Prandtl number and the relaxation times of rotational and vibrational energies. With the reduced distribution technique , this model can be reduced to a three distribution system that could be used to simulate polyatomic gases with rotational and vibrational energy for a computational cost close to that of a simple monoatomic gas.
Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation
本文提出一种新的隐式半拉格朗日方案，用于椭球形BGK模型，通过利用椭球形高斯的特殊结构，将其转化为半显式形式，同时保证隐式方法的稳定性与显式方法的效率，并在加权$ L^{\infty} $范数下推导了误差估计。
Ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a Prandtl parameter $ν$, so that the correct transport coefficients can be computed in the Navier-Stokes limit. In this work, we consider the existence and uniqueness of stationary solutions for the ES-BGK model in a slab imposed with the mixed boundary conditions. One of the key difficulties arise in the uniform control of the temperature tensor from below. In the non-critical case $(-1/2<ν<1)$, we utilize t…
Unique mild solution for the ES-BGK model with the correct Prandtl number
该研究针对具有正确普朗特数的ES-BGK模型，通过推导各向异性速度矩估计，结合均匀椭圆性等条件，证明了其温和解的存在性与唯一性。
Ellipsoidal BGK model near a global Maxwellian
本文研究了椭圆BGK模型在初始数据为全球麦克斯韦分布小扰动时经典解的存在性，通过分析发现该模型退化程度与原始BGK模型或玻尔兹曼方程在-1/2<ν<1范围内相似。
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