A Revisit of the Velocity Averaging Lemma: On the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain

Chen, I-Kun; Chuang, Ping-Han; Hsia, Chun-Hsiung; Su, Jhe-Kuan

doi:10.1007/s10955-022-02977-5

Cited by 2 publications

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References 25 publications

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“…In [11,3], they use a kinetic distance function and prove a weighted estimate beyond H 1 x . In [6,5], other authors later show stationary solutions are in H 1− x , namely a fractional Sobolev space in space of order strictly less than 1. All these studies left the H 1 threshold problem unsolved.…”

Section: Introductionmentioning

confidence: 94%

Cercignani-Lampis boundary in the Boltzmann theory

Chen¹

2020

Kinetic &Amp; Related Models

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The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an L ∞ estimate. In particular, for the L ∞ bound we develop a new decomposition on the boundary term combining with repeated interaction through the characteristic. Via this method, we construct a unique steady solution of the Boltzmann equation with constraints on the wall temperature and the accommodation coefficient.

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Section: Introductionmentioning

confidence: 94%

Cercignani-Lampis boundary in the Boltzmann theory

Chen¹

2020

Kinetic &Amp; Related Models

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A remark on the velocity averaging lemma of the transport equation with general case

Lyu,

Sun

2024

NHM

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<abstract>In this paper, we proved a new result for the celebrated velocity averaging lemma of the free transport equation with general case <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \partial_{t}f+ a(v) \cdot \nabla_{x} f = 0\,. \end{equation*} $\end{document} </tex-math></disp-formula> After averaging with some weight functions $ \varphi(v) $, we proved that the average quantity $ \rho_{\varphi}(t, x) = \int_{\mathbb{R}_{v}^{3}}f(t, x, v)\, \varphi(v)\, {\rm d} v $ is in $ W_{x}^{1, p} $, $ p\in[1, +\infty] $. This result revealed the regularizing effect for the mean value with respect to the velocity of the solution. Our strategy was taking advantage of a modified vector field method to build up a bridge between the $ x $-derivative and $ v $-derivative. One significant point was that we first observed that the operator $ t\, \nabla_{x}+\left(\left[ \nabla _{v} a(v) \right] ^{T}\right) ^{-1}\nabla_{v} $ commuted with $ \partial_{t}+ a(v) \cdot \nabla_{x} $.</abstract>

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A Revisit of the Velocity Averaging Lemma: On the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain

Cited by 2 publications

References 25 publications

Cercignani-Lampis boundary in the Boltzmann theory

Cercignani-Lampis boundary in the Boltzmann theory

A remark on the velocity averaging lemma of the transport equation with general case

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