Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

Guo, Yan; Huang, Feimin; Wang, Yong

doi:10.1007/s00205-021-01651-6

Cited by 26 publications

(11 citation statements)

References 45 publications

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“…This framework was extended to treat the Vlasov-Poisson-Boltzmann system in Guo-Jang [20] and the relativistic Boltzmann equation in Speck-Strain [44]. Recently, this framework was further developed to the investigation of the relativistic Vlasov-Maxwellian-Boltzmann system in Guo-Xiao [22] and the Boltzmann equation with boundary conditions in half-space in Guo-Huang-Wang [18], Jiang-Luo-Tang [32,31].…”

Section: Background and Literature Reviewmentioning

confidence: 99%

Hilbert Expansion for the Relativistic Landau Equation

Ouyang¹,

Wu²,

Xiao³

2022

Preprint

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In this paper, we study the local-in-time validity of the Hilbert expansion for the relativistic Landau equation. We justify that solutions of the relativistic Landau equation converge to small classical solutions of the limiting relativistic Euler equations as the Knudsen number shrinks to zero in a weighted Sobolev space. The key difficulty comes from the temporal and spatial derivatives of the local Maxwellian, which produce momentum growth terms and are uncontrollable by the standard L 2 -based energy and dissipation. We introduce novel time-dependent weight functions to generate additional dissipation terms to suppress the large momentum. The argument relies on a hierarchy of energy-dissipation structures with or without weights. As far as the authors are aware of, this is the first result of the Hilbert expansion for the Landau-type equation.

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Section: Background and Literature Reviewmentioning

confidence: 99%

Hilbert Expansion for the Relativistic Landau Equation

Ouyang¹,

Wu²,

Xiao³

2022

Preprint

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“…for any |β| = m ≥ 0. By the existence theory of the system (1.31) (see [10,12], for instance), the all quantities h n m (n ≥ 1, m ≥ 0) can be bounded by the quantity e m ′ =…”

Section: Applications To Hilbert Expansions Of Boltzmann Equation Wit...mentioning

confidence: 99%

“…One of the goals of this paper is to investigate the decay (with respect to v) estimates of the pseudo-inverse operator L −1 . This decay estimate has been widely applied to hydrodynamic limits of the Boltzmann equations (for examples, [7,9,10]). Specifically, in any fluid limits of kinetic equations, once the Hilbert expansion approach is employed, the solution will be formally written as F κ = ∞ n=0 κ n F n , where the kinetic part of F n is solved from LF n = G n .…”

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confidence: 99%

Grad-Caflisch type decay estimates of pseudo-inverse of linearized Boltzmann operator and application to Hilbert expansion of compressible Euler scaling

Ning¹,

Luo²,

Tang³

2022

Preprint

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In this paper, we prove some Grad-Caflisch type decay estimates of the pseudoinverse of linearized Boltzmann collision operator, including both the hard potential (0 ≤ γ ≤ 1) and part of soft (− 3 2 < γ < 0) potential cutoff interaction kernels. The key idea is that the weighted L ∞ -norms of (L − ν)f are first dominated by the weighted L 2 -norms of f , and then the L 2 -norms are bounded by the L ∞ -norms of Lf via the hypocoercivity of the weighted operator L. The proof of the weighted hypocoercivity employs the high-low velocities estimates argument. Finally, these decay estimates are further applied to derive some new point-wise estimates for the Hilbert expansion terms of the Boltzmann equation in the compressible Euler scaling.

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“…Following the Caflisch's strategy together with a new L 2 − L ∞ approach developed in [28], Guo-Jang-Jiang [32,33] removed the restriction with zero initial condition so that the positivity of the Boltzmann solution can be guaranteed. Recently, Guo-Huang-Wang [29] and Jiang-Luo-Tang [42,43] made significant progress on the topic by making use of the L 2 − L ∞ interplay technique to generalize the results of [6,32,33] to the half-space problem with different types of boundary conditions including specular reflection, diffuse reflection and even the mixed Maxwell reflection. We also mention another recent progress Guo-Xiao [34] for an application of the L 2 − L ∞ approach to the global Hilbert expansion to the relativistic Vlasov-Maxwell-Boltzmann system.…”

Section: Introductionmentioning

confidence: 99%

Compressible fluid limit for smooth solutions to the Landau equation

Duan¹,

Chen²,

Yu³

2022

Preprint

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Although the compressible fluid limit of the Boltzmann equation with cutoff has been well investigated in [6] and [33], it still remains largely open to obtain analogous results in case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, essentially due to the velocity diffusion effect of collision operator such that L ∞ estimates are hard to obtain without using Sobolev embeddings. In the paper, we are concerned with the compressible Euler and acoustic limits of the Landau equation for Coulomb potentials in the whole space. Specifically, over any finite time interval where the full compressible Euler system admits a smooth solution around constant states, we construct a unique solution in a high-order weighted Sobolev space for the Landau equation with suitable initial data and also show the uniform estimates independent of the small Knudsen number ε > 0, yielding the O(ε) convergence of the Landau solution to the local Maxwellian whose fluid quantities are the given Euler solution. Moreover, the acoustic limit for smooth solutions to the Landau equation in an optimal scaling is also established. For the proof, by using the macro-micro decomposition around local Maxwellians together with techniques for viscous compressible fluid and properties of Burnett functions, we design an ε-dependent energy functional to capture the dissipation in the compressible fluid limit with feature that only the highest order derivatives are most singular.

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Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

Cited by 26 publications

References 45 publications

Hilbert Expansion for the Relativistic Landau Equation

Hilbert Expansion for the Relativistic Landau Equation

Grad-Caflisch type decay estimates of pseudo-inverse of linearized Boltzmann operator and application to Hilbert expansion of compressible Euler scaling

Compressible fluid limit for smooth solutions to the Landau equation

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