Equilibrating Effects of Boundary and Collision in Rarefied Gases

Kuo, Hung Wen; Liu, Tai Ping; Tsai, Li Cheng

doi:10.1007/s00220-014-2042-9

Cited by 15 publications

(30 citation statements)

References 10 publications

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“…in space dimension d = 1, 2. This allows us to use the stochastic formulation of our previous works [10] and [11], which provides an explicit description of the evolution of the free molecular flow. We decompose the microscopic velocity ζ = (ζ 1 , ζ 2 , ζ 3 ) ∈ R 3 into…”

Section: Rtmentioning

confidence: 99%

“…The present study focuses first on the equilibrating effect of the boundary condition by considering the free molecular flows. This approach has been considered for the diffuse reflection boundary condition, initiated by [16] for one space dimension and then generalized to higher space dimensions by [10] for constant boundary temperature, and by [11] for variable boundary temperature. It also has been studied for half space under gravitational force [12].…”

Section: Theorem 15 (Existence Of Steady Solution For Full Boltzmannmentioning

confidence: 99%

“…Remark 2.4. When α = 1, the case of diffuse reflection boundary condition, we have the convergence rate (1 + t) −d of the boundary flux, [10,11]. Roughly speaking, the equilibrating effect is mainly from sufficiently many collisions with the boundary of diffuse reflection condition when t is large.…”

Section: 1mentioning

confidence: 99%

“…We achieve the aim through (44) and (45). For the events that diffusion reflections are more than specular reflections, we may modify the analysis from the previous works [10] and [11] to get the decay rate, (48). However, the appearance of specular reflection will slow down the decay rate.…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

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Equilibrating Effect of Maxwell-Type Boundary Condition in Highly Rarefied Gas

Kuo

2015

J Stat Phys

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We study the equilibrating effects of the boundary and intermolecular collision in the kinetic theory for rarefied gases. We consider the Maxwell-type boundary condition, which has weaker equilibrating effect than the commonly studied diffuse reflection boundary condition. The gas region is the spherical domain in R d , d = 1, 2. First, without the equilibrating effect of the collision, we obtain the algebraic convergence rates to the steady state of free molecular flow with variable boundary temperature. The convergence behavior has intricate dependence on the accommodation coefficient of the Maxwell-type boundary condition. Then we couple the boundary effect with the intermolecular collision and study their interaction. We are able to construct the steady state solutions of the full Boltzmann equation for large Knudsen numbers and small boundary temperature variation. We also establish the nonlinear stability with exponential rate of the stationary Boltzmann solutions. Our analysis is based on the explicit formulations of the boundary condition for symmetric domains.

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Section: Rtmentioning

confidence: 99%

Section: Theorem 15 (Existence Of Steady Solution For Full Boltzmannmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Proof Of Theorem 22mentioning

confidence: 99%

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Equilibrating Effect of Maxwell-Type Boundary Condition in Highly Rarefied Gas

Kuo

2015

J Stat Phys

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“…Green's function approach has been useful for the study of nonlinear waves for viscous conservation laws [4,22,23,45,46]. Green's function approach [32,34,47,48] for the Boltzmann equation has yielded quantitative understanding of nonlinear waves and the boundary behaviour [41,[49][50][51]. The relation between the gas dynamics and the kinetic theory is a rich field.…”

Section: Discussionmentioning

confidence: 99%

Aspects of dissipation for compressible fluids and kinetic theory

Liu

2013

Phil. Trans. R. Soc. A.

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There are several types of dissipation—the viscosity and heat conductivity, the nonlinearity and the coupling of distinct characteristics—that can occur in the system of conservation laws. In addition to these corresponding types of dissipation, there are other consequences of the H-theorem and the boundary dissipative effects for the kinetic theory. We discuss these issues and raise questions.

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The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

Liu¹,

Yang²

2016

Arch Rational Mech Anal

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Abstract. Boundary effects are central to the dynamics of the dilute particles governed by Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for Boltzmann equation with soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate L ∞ analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in L ∞ space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori L ∞ estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the L 2 − L ∞ theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted L ∞ theory so that we could deal with the greater difficulties stemmed from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in L ∞ space with the aid of the L 2 theory and a bootstrap argument. These methods in the latter case can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.

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Equilibrating Effects of Boundary and Collision in Rarefied Gases

Cited by 15 publications

References 10 publications

Equilibrating Effect of Maxwell-Type Boundary Condition in Highly Rarefied Gas

Equilibrating Effect of Maxwell-Type Boundary Condition in Highly Rarefied Gas

Aspects of dissipation for compressible fluids and kinetic theory

The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

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