Free Molecular Flow with Boundary Effect

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

doi:10.1007/s00220-013-1662-9

Cited by 22 publications

(38 citation statements)

References 9 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%

“…Therefore, we will consider the initial-boundary value problem of equation (9) with boundary condition (10) and initial data:…”

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%

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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%

“…Therefore, we will consider the initial-boundary value problem of equation (9) with boundary condition (10) and initial data:…”

Section: Rtmentioning

confidence: 99%

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

confidence: 99%

Section: 1mentioning

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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“…It is a priori unclear whether we can reach the limit rate O 1 √ t or can go beyond this rate for the kinetic semigroups e tTO t≥0 (note that much better rates of convergence occur for bounded initial datum in balls, see [1] [15]). We refer to Remark 31 and Remark 32 for different open problems suggested by our construction.…”

Section: Introductionmentioning

confidence: 99%