Stationary Quantum BGK Model for Bosons and Fermions in a Bounded Interval

Bae, Gi-Chan; Yun, Seok-Bae

doi:10.1007/s10955-019-02466-2

Cited by 8 publications

(5 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The existence of unique mild solutions were obtained in [7], using classical Banach fixed point argument. This argument were then applied to the relativistic BGK model of Marle type [26] and to the quantum BGK model for non-saturated Fermion system and the Boson system without condensation [6].…”

Section: Resultsmentioning

confidence: 99%

Stationary BGK models for chemically reacting gas in a slab

Kim,

Lee,

Yun

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We study the boundary value problem of two stationary BGK-type models -the BGK model for fast chemical reaction and the BGK model for slow chemical reaction -and provide a unified argument to establish the existence and uniqueness of stationary flows of reactive BGK models in a slab. For both models, the main diffculty arises in the uniform control of the reactive parameters from above and below, since, unlike the BGK models for non-reactive gases, the reactive parameters for the reactive BGK models are defined through highly nonlinear relations. To overcome this difficulty, we introduce several nonlinear functionals that capture essential structures of such nonlinear relations such as the monotonicity in specific variables, that enable one to derive necessary estimates for the reactive equilibrium coefficients.

show abstract

Section: Resultsmentioning

confidence: 99%

Stationary BGK models for chemically reacting gas in a slab

Kim,

Lee,

Yun

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Proof of Theorem 2.1 (1), (2). The proof for (1) can be found in [3]. Therefore, we start with the proof of (2).…”

Section: 4mentioning

confidence: 99%

“…More precisely, whether the relaxation operator can be soundly defined in a rigorous manner so that it satisfies the same conservation laws and the H-theorem as the quantum Boltzmann has never been rigorously verified in the literature. The well-definedness of such equilibrium coefficients for M 11 and M 22 follows directly from the relevant results for the one-species quantum BGK model in [3,4,21,42,47]. Thus, we focus on the determination of the equilibrium coefficients for the mixture equilibrium M 12 and M 21 .…”

mentioning

confidence: 97%

BGK model of the multi-species Uehling-Uhlenbeck equation

Bae¹,

Klingenberg²,

Pirner³

et al. 2021

KRM

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In [9], however, the boundary condition was limited to inflow boundary condition, and the case ν = −1/2 is not treated, which is the main motivation of the current work. The argument of [9] was then applied to a relativistic BGK model [30] and to the quantum BGK model [8].…”

Section: Theorem 14 [Diffusive Dominant Case]mentioning

confidence: 99%

Stationary Flows of the ES-BGK model with the correct Prandtl number

Brull,

Yun

2020

Preprint

Self Cite

View full text Add to dashboard Cite

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 the property that the temperature tensor is equivalent to the temperature in this range. In the critical case, (ν = −1/2), where such equivalence relation breaks down, we observe that the size of bulk velocity in x direction can be controlled by the discrepancy of boundary flux, which enables one to bound the temperature tensor from below.

show abstract

Stationary Quantum BGK Model for Bosons and Fermions in a Bounded Interval

Cited by 8 publications

References 44 publications

Stationary BGK models for chemically reacting gas in a slab

Stationary BGK models for chemically reacting gas in a slab

BGK model of the multi-species Uehling-Uhlenbeck equation

Stationary Flows of the ES-BGK model with the correct Prandtl number

Contact Info

Product

Resources

About