Existence, uniqueness and positivity of solutions for BGK models for mixtures

Klingenberg, Christian; Pirner, Marlies

doi:10.1016/j.jde.2017.09.019

Cited by 29 publications

(33 citation statements)

References 24 publications

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“…Examples include the model of Gross and Krook [29], the model of Hamel [32], the model of Greene [27], the model of Garzo, Santos and Brey [26], the model of Sofonea and Sekerka [57], the model by Andries, Aoki and Perthame [1], the model of Brull, Pavan and Schneider [13], the model of Klingenberg, Pirner and Puppo [40], the model of Haack, Hauck, Murillo [31] and the model of Bobylev, Bisi, Groppi, Spiga [10]. BGK models have also been extended to ES-BGK models, polyatomic molecules or chemically reactive gas mixtures; see for example [8,9,14,28,38,39,41,49,60]. BGK models are often used in applications because they give rise to efficient numerical computations as compared to models with Boltzmann collision terms [5,6,16,17,24,48,53,54].…”

Section: 4mentioning

confidence: 99%

BGK model of the multi-species Uehling-Uhlenbeck equation

Bae¹,

Klingenberg²,

Pirner³

et al. 2021

KRM

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We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and H-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficients of the local equilibriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under the condition, leading to the well-definedness of our model.

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Section: 4mentioning

confidence: 99%

BGK model of the multi-species Uehling-Uhlenbeck equation

Bae¹,

Klingenberg²,

Pirner³

et al. 2021

KRM

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“…Perthame and Pulvirenti later studied the existence of unique mild solution in a weighted L ∞ space [31]. These works were fruitfully extended to several directions: gases in the presence of external forces [43], plasma [50,51], solutions in L p spaces [49], ellipsoidal extension [47] and gas mixture problems [23]. The existence of classical solutions and its exponential stabilization near equilibrium are studied in [46,48].…”

Section: Introductionmentioning

confidence: 99%

Global existence of weak solutions for Navier–Stokes-BGK system

Choi

Yun

2020

Nonlinearity

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In this paper, we study the global well-posedness of a coupled system of kinetic and fluid equations. More precisely, we establish the global existence of weak solutions for Navier-Stokes-BGK system consisting of the BGK model of Boltzmann equation and incompressible Navier-Stokes equations coupled through a drag forcing term. This is achieved by combining weak compactness of the particle interaction operator based on Dunford-Pettis theorem, strong compactness of macroscopic fields of the kinetic part relied on velocity averaging lemma and a high order moment estimate, and strong compactness of the fluid part by Aubin-Lions lemma.

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“…[9][10][11][12] and multirelaxation models, e.g. [13][14][15] . A new consistent BGK mixture model was introduced recently 16 for both Maxwellian and arbitrary potential molecules.…”

Section: Introductionmentioning

confidence: 99%

Numerical evaluation of novel kinetic models for binary gas mixture flows

2020

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Most flows of practical interest consist of mixture of gases. Therefore the capability to model a gas mixture flow is important. Kinetic models for multicomponent gases have been considered since the original Bhatnagar-Gross-Krook (BGK) model was formulated. BGK-derived models pose a number of difficulties, e.g. avoiding negative density and temperature(s). A distinct challenge of the BGK approximation lies in recovering correct transport coefficients in the continuum limit. Two new kinetic models for gas mixtures: a Shakhov-based model and an Ellipsoidal-Statistical (ES)-based model, were recently introduced. Both models are capable of modelling a binary mixture of monoatomic gases and account for separate species-mean velocity such that the species diffusion and velocity drift are accurately represented. The main advantage is the recovery of three correct transport coefficients in the hydrodynamic limit and as a result having a correct Prandtl number for the mixture. The goal of this paper is to numerically validate the two new kinetic models for a range of high-speed flows and demonstrate their capabilities and limitations. The models are first validated against known results for normal shocks, showing good agreement for species density and temperature profiles. Moreover, the importance of the Prandtl number correction is demonstrated with the evaluation of the heat flux. A parametric study demonstrates the variation in flow properties for different mass ratios between species and for different Mach numbers. Finally, the models are evaluated for the flow around a circular cylinder. A detailed comparison with Monte Carlo results demonstrates promising results from both kinetic models.

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Existence, uniqueness and positivity of solutions for BGK models for mixtures

Cited by 29 publications

References 24 publications

BGK model of the multi-species Uehling-Uhlenbeck equation

BGK model of the multi-species Uehling-Uhlenbeck equation

Global existence of weak solutions for Navier–Stokes-BGK system

Numerical evaluation of novel kinetic models for binary gas mixture flows

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