Singularity of Macroscopic Variables Near Boundary for Gases with Cutoff Hard Potential

Chen, I-Kun; Hsia, Chun-Hsiung

doi:10.1137/140986220

Cited by 6 publications

(14 citation statements)

References 20 publications

(6 reference statements)

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“…Here, the constants 0 < ν 0 < ν 1 may depend on the potential and C 1 and C 2 may depend on δ and the potential. Notice that (2.5) was established in [1] and (2.6) can be concluded by the observation in [6] in case the cross section satisfies (1.7).…”

Section: Properties Of Linearized Collision Operatormentioning

confidence: 76%

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Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

Chen¹,

Hsia²,

Kawagoe³

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma. 4 3 +ǫ . Now we briefly recall some ideas about the velocity averaging effect for the stationary linearized Boltzmann equation on bounded convex domains 9

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Section: Properties Of Linearized Collision Operatormentioning

confidence: 76%

“…Then, by Schur's test, we can conclude the following smooth effect of K in velocity variable mentioned in [6].…”

Section: Properties Of Linearized Collision Operatormentioning

confidence: 93%

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

Chen¹,

Hsia²,

Kawagoe³

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…For the regularity in space, the Hölder continuity of K(f ) in space for solutions to one space dimensional equations is proved in [2] for hard sphere gases, again thanks to the simple geometry of one space dimension. This observation also plays an important role in [4]. More precisely, in one space dimensional case, any two points can be connected by a trajectory, either forward or backward, for almost all velocity.…”

Section: Introductionmentioning

confidence: 90%

“…In [2], the locally Hölder continuity of K(f ) in velocity has been shown for f ∈ L 2 ζ for hard sphere gases. Here, we will use an estimate appeared in [4].…”

Section: Gaining Regularity From Collision and Transportmentioning

confidence: 99%

“…This cutoff was first introduced by Grad [9], and later the analysis was refined by Caflisch [1]. Furthermore, we assume the cross section is a product of a function of the length of relative velocity and a function of the deflecting angle as in [4], i.e., we assume that the cross section is nontrivial and satisfies B(|ζ * − ζ|, θ) = |ζ * − ζ| γ β(θ), 0 ≤ β(θ) ≤ C cos θ sin θ. before linearized around the standard Maxwellian π − 3 2 e −|ζ| 2 . Under this assumption, L has the following known properties (See [1,4,9]).…”

Section: Introductionmentioning

confidence: 99%

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Regularity of Stationary Solutions to the Linearized Boltzmann Equations

Chen¹

2018

SIAM J. Math. Anal.

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We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded C 1 convex domains in R 3 for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary solutions solutions are Hölder continuous with order 1 2 − away from the boundary provided the incoming data have the same regularity. The key idea is to partially transfer the regularity in velocity obtained by collision to space through transport and collision.

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Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

Takata

Hattori

2022

J Stat Phys

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Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.

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Singularity of Macroscopic Variables Near Boundary for Gases with Cutoff Hard Potential

Cited by 6 publications

References 20 publications

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

Regularity of Stationary Solutions to the Linearized Boltzmann Equations

Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

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