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Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits

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Pith's one-line read For intermediate Maxwell accommodation, the half-space Boltzmann Knudsen layer problem is well-posed in weighted $L^\infty_{x,v}$, with exponential decay and a uniquely determined far-field state.

desk verdict Fills the missing 0<α<1 case for the Knudsen layer with general sources; the proof leans on an omitted estimate that a referee must check before the main theorem can be trusted. read the letter →

arxiv 2501.08733 v1 pith:FAJXEK6E submitted 2025-01-15 math.AP

Ling-Bing He , Ning Jiang , Yulong Wu This is my paper · ORCID
classification math.AP MSC 35Q2076P0535F3035A0135A02
keywords KnudsenlayerequationBoltzmannMaxwellreflectionboundaryconditionaccommodationcoefficientslipconditionshydrodynamiclimitwell-posednesshard-spherecollisions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes well-posedness of the half-space Knudsen layer equation for the Boltzmann equation when the wall reflects particles by Maxwell's mixed law, with accommodation coefficient strictly between $0$ and $1$. The central theorem states that for hard-sphere collisions and source terms satisfying a natural moment condition, the nonlinear layer problem has exactly one solution in a weighted $L^\infty_{x,v}$ space, that the solution decays exponentially in the distance from the wall, and that its far-field limit is uniquely determined and lies in a three-dimensional subspace of the collision null space. This closes the previously open middle range of accommodation coefficients, complementing existing results for purely specular ($\alpha=0$) and purely diffuse ($\alpha=1$) reflection. The result matters because the Knudsen layer determines the slip boundary conditions used in hydrodynamic limits: the paper derives, for a linearized incompressible Navier-Stokes limit, first-order slip conditions with slip coefficients depending on $\alpha$, and it gives an explicit description of the 'vanishing sources set' from earlier work.

What carries the argument

The load-bearing device is an auxiliary system with prescribed incoming mass flux. The problem (1.12) with the far-field condition is overdetermined, so the authors first solve (1.25), where the outgoing flux is fixed by $P_\gamma f(0)=\lambda$; the diffuse part of the boundary condition is treated as a source, and standard characteristic estimates apply. They solve this on a finite slab $x\in(0,d)$ with specular reflection at $x=d$, adding an artificial damping $\epsilon f$, constructing solutions by a contraction argument in the accommodation coefficient factor, then passing $\epsilon\to 0$ and $d\to\infty$. A separate uniqueness argument shows the half-space solution is independent of the auxiliary parameter $\lambda$. In the fluid-limit application, the second mechanism is the symmetry reduction: for functions of the form $v_i\varphi(|v|,v_3)$, the operator $L$ acts as a scalar operator $L_S$ whose null space on the reduced measure is just $\mathrm{span}\{\sqrt{m}\}$, which lets the tangential and thermal layer profiles be solved separately and the slip coefficients be read off.

What would settle it

Run the finite-slab iteration (2.11) with zero source and zero data except a small flux $\lambda$, for several $\alpha\in(0,1)$, and check the claimed bound (2.5) uniform in $\epsilon$ and $a$; a single configuration where the weighted $L^\infty$ norm grows without control as $d$ increases would disprove the theorem.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is Theorem 1.2: under the solvability condition (1.18), the nonlinear Knudsen layer problem (1.21) has a unique solution $f$ with $\|e^{\sigma x}wf\|_{L^\infty}+|wf(0)|_{L^\infty(\gamma)}<\infty$, and the far-field $q_\infty$ is uniquely determined with $q_\infty=(b^\infty_1 v_1+b^\infty_2 v_2+c^\infty(|v|^2-3)/2)\sqrt{m}$. The linear version, Theorem 1.1, proves the same for (1.12) and identifies the admissible far-field state rather than imposing vanishing at infinity. In Section 4 the theorem is applied to the linearized incompressible Navier-Stokes limit: solving the resulting layer problems and using rotational symmetry of $L$ and of the boundary operator $L_R-\alpha L_D$ yields the slip boundary conditions $u_{1,i}=c_u(\partial_3 u_{0,i}+\partial_i u_{0,3})$ and $\theta_1=c_\theta\partial_3\theta_0$, with the slip coefficients $c_u,c_\theta$ determined by the accommodation coefficient.

Load-bearing premise

The proof rests on Lemma 2.1, a weighted $L^\infty$ a priori estimate for the finite-slab problem with mixed specular-diffuse boundary, whose proof is omitted and merely said to follow Lemma 3.3 of [15]; if that estimate does not actually transfer to the case where the diffuse part is treated as a source, the existence argument in Section 2 collapses.

Editorial extensions

If this is right

  • The vanishing sources set introduced in [18] is now explicitly characterized: for general sources satisfying the solvability condition (1.18), the layer solution tends to a uniquely determined far-field $q_\infty$ in $\mathrm{span}\{v_1\sqrt{m},v_2\sqrt{m},(|v|^2-3)\sqrt{m}/2\}$.
  • For the linearized incompressible Navier-Stokes limit, the first-order slip conditions are $u_{1,i}=c_u(\partial_3 u_{0,i}+\partial_i u_{0,3})$ and $\theta_1=c_\theta\partial_3\theta_0$, where $c_u$ and $c_\theta$ are determined by the accommodation coefficient $\alpha$; these are genuine mixed-reflection slip laws, different from the specular and almost-specular cases.
  • The method applies to other hydrodynamic limits, including incompressible Euler and compressible Euler, because the nonlinear term $\Gamma$ has the same rotational symmetry used to separate layer profiles.
  • For every $\alpha\in(0,1)$ the boundary layer decays exponentially with rate $\sigma$, so matched-asymptotic expansions can match the layer to the fluid interior without artificial far-field cutoffs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The auxiliary-mass-flux device is likely portable to other kinetic boundary conditions, such as Cercignani-Lampis or partial accommodation with velocity-dependent coefficients, since it separates the overdetermination issue from the specific form of the reflection law.
  • A concrete numerical test of the predicted slip coefficients is available: solve the one-dimensional layer problems (4.13) and (4.14) for hard spheres and compare $c_u(\alpha)$ and $c_\theta(\alpha)$ with kinetic-theory tables; agreement would confirm the theorem's application, and disagreement would point to the omitted estimate in Lemma 2.1.
  • The exponential decay estimate suggests that the layer has a finite effective thickness controlled by $\sigma_0-\sigma$; if the strategy extends beyond hard spheres, the same uniqueness of $q_\infty$ would supply slip conditions for more general collision kernels.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper studies the steady Boltzmann equation in the half-space, linearized around a global Maxwellian with hard-sphere collisions and Maxwell reflection boundary conditions with accommodation coefficient 0<α<1. It states a linear theorem (Theorem 1.1) giving existence, uniqueness, exponential decay, and uniqueness of the far-field state for source terms satisfying a solvability condition, and a nonlinear theorem (Theorem 1.2) for the quadratic nonlinearity under a smallness assumption. The proof proceeds through an auxiliary finite-slab problem with prescribed mass flux, artificial damping, and contraction and compactness arguments in weighted L∞ and L2. Section 4 applies the linear theorem to the linearized incompressible Navier-Stokes limit and sketches the derivation of slip boundary conditions. The main theorem is plausible and would fill the gap 0<α<1 in the weighted L∞ framework, but several load-bearing estimates are asserted without proof, most notably the weighted L∞ a priori estimate in Lemma 2.1 and the d→∞ passage in Section 2.4.

Significance. If the central theorems and omitted estimates are fully established, the paper would provide a rigorous well-posedness theory for the Boltzmann Knudsen layer with Maxwell reflection for all non-endpoint accommodation coefficients, extending the endpoint cases treated in [14,15] and complementing the vanishing-sources approach of [18]. The nonlinear iteration via contraction is a natural and credible route, and the regularity framework with a Gaussian-then-algebraic weight is appropriate for hard-sphere collisions with angular cutoff. The application to fluid limits is more modest than the abstract suggests: the paper establishes an existence and uniqueness statement for the layer and represents the slip coefficients as determined constants, but it does not compute or explicitly derive their values. The contribution is significant if the missing proofs are supplied, but the manuscript in its current form is not self-contained.

major comments (5)
  1. [§2.1, Lemma 2.1, estimate (2.5)] Lemma 2.1 is the sole weighted L∞ a priori estimate for the finite-slab problem with the mixed Maxwell boundary, and its proof is omitted: the text says it 'closely follows Lemma 3.3 in [15]' and gives no details. This estimate is then used in Lemma 2.3 to close the contraction, in Lemma 2.5 for the ε→0 limit, and in Corollary 2.1 and Lemma 2.8 for the d→∞ passage. If Lemma 3.3 of [15] does not transfer to the mixed boundary with the diffuse part treated as a source term w(r̃), the existence proof for the auxiliary system (1.25), and hence Theorems 1.1 and 1.2, has no replacement support. Please provide the full proof, or give a precise statement and proof of the transfer from [15] to the boundary condition in (2.3), including the treatment of the diffuse source term and the weight w.
  2. [§2.4, estimates (2.73)–(2.74)] The d→∞ limit from (1.26) to (1.25) is the step that actually constructs the half-space solution in Lemma 1.1, yet the two displayed L2 and L∞ estimates are justified only by 'The proof is similar so we omit the details here for brevity'. These estimates are load-bearing: they provide the Cauchy property of the sequence (f̃_d, q(d)) and are used to obtain the bounds (2.75) and (2.78) that enter Theorem 1.1. Please include the derivation of (2.73)–(2.74), or at least explicitly identify the arguments in Section 3.3 of [15] and state the modifications needed for the mixed boundary condition at x=0.
  3. [§2.2, Lemma 2.4] The proof of Lemma 2.4 displays the estimate for cε in (2.29), but the corresponding estimates for bε and aε are dismissed with 'The proof is similar, so we omit the details'. Since the uniform-in-ε bound for the full macroscopic projection Pfε is needed in Lemma 2.5 to pass ε→0, the missing bε and aε estimates should either be written out with their test functions or referenced to a specific result that covers the Maxwell boundary with 0<α<1. Without these bounds, the passage from (1.27) to (2.30) is incomplete.
  4. [§4, Lemmas 4.1–4.2 and equations (4.11)–(4.14)] The application to fluid limits stops short of deriving the slip boundary conditions stated in the abstract. Lemmas 4.1 and 4.2 provide existence and bounds for the pairs (cθ,φθ) and (cu,φu), but Lemma 4.2 is justified only by 'similar arguments' plus citations to [2,6], and the final boundary conditions display cu and cθ as undetermined constants, with the text referring to [1,13,24,25] for their values. The paper should either present explicit formulas or numerical data for cu and cθ in the hard-sphere Maxwell case, or explicitly describe the contribution as a rigorous reduction of the slip problem to the solvability of (4.13)–(4.14) rather than as a derivation of the slip coefficients.
  5. [Remark 1.1 and Introduction, vanishing sources set] The paper states that Theorem 1.1 'explicitly characterizes the vanishing sources set' introduced in [18], but the theorem only associates to each admissible pair (g,r) a unique far-field state q∞; it does not give an explicit condition for q∞=0, which is what would characterize the vanishing sources set as a set. Please either formulate VSS as an effectively checkable set in terms of (g,r), or weaken the claim to a characterization of the mapping (g,r)↦q∞.
minor comments (4)
  1. [Abstract and throughout] There are numerous typographical errors, including 'expample' in the abstract, 'existance' in the Section 2.1 heading, 'deontes' in Section 1.4, 'Approxiamte' in Section 2.1, and 'arguements' in several places; a careful proofreading pass is recommended.
  2. [Theorem 1.1, estimates (1.15)–(1.17)] The notation for the boundary norm of r is inconsistent: (1.15) uses |wr|_{L∞_v(R3_+)}, while (1.16)–(1.17) use |wr|_{L∞(γ−)}; please unify these notations and state clearly on which side of the boundary the norm is taken.
  3. [§4, Lemmas 4.1–4.2] The statement 'α=O(1), α∈(0,1]' conflates the main theorem's assumption 0<α<1 with the endpoint α=1 treated by Theorem 1.5 in [14]; the text should explicitly note that the endpoint is handled by a separate theorem and not by Theorem 1.1.
  4. [§4, proof of Lemma 4.1] The uniqueness argument cites 'Lemma 2.1.1 in [10]' for the conclusion Φ≡0, but the paper earlier refers to the same kind of statement as 'Lemma 2.1.1 of [6]'; please verify the reference and use a consistent citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the central existence results are derived from auxiliary-system estimates and contraction arguments, with the only notable gap being an externally cited a priori estimate that is a support gap rather than a circular step.

full rationale

Walking the derivation chain: Theorem 1.2 is obtained by iterating the linear Theorem 1.1 on the quadratic term Γ(f_i,f_i), with the estimate of Γ quoted from Guo [12] and a smallness condition closing the contraction; it does not assume the nonlinear solution. Theorem 1.1 is obtained from Lemma 1.1, the well-posedness of the auxiliary system with prescribed mass flux, followed by the λ-independence argument in Section 3.1. Lemma 1.1 is obtained from finite-slab a priori estimates (Lemmas 2.1–2.5), the passage ǫ→0, the passage d→∞ (Lemmas 2.6–2.10), and a separate uniqueness argument; none of these steps uses the theorem being proved as an input. The application in Section 4 uses Theorem 1.1 on the reduced boundary-value problem and cites [1,13,24,25] only for the numerical/kinetic values of the slip coefficients; the constants c_u and c_theta are solved together with φ_u and φ_theta from the boundary-layer problems, not fitted from the theorem's output. The self-citations [16]–[19] supply background and the VSS terminology, and the VSS remark is a descriptive interpretation of Theorem 1.1, not evidence for it. The one genuine weakness, flagged in Section 2.1, is that the proof of Lemma 2.1 is omitted and deferred to Lemma 3.3 of [15]; this is a support gap and a correctness risk for the mixed-boundary adaptation, but it is not circularity, because the estimate is an external input and the conclusion does not reduce to that estimate by definition or by a self-citation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its results rest on standard kinetic theory: the hard-sphere collision kernel, the spectral theory of the linearized collision operator, and two external lemmas from earlier boundary-layer papers that are used without proof. The main theorem is a new existence and uniqueness statement, not a consequence of those lemmas alone.

assumptions (4)
  • domain assumption Hard-sphere collision kernel with angular cutoff, b(|v*−v|,ω)=|(v*−v)·ω|.
    Invoked in Section 1.1; gives collision frequency ν∼1+|v| and compactness of K, both used throughout Sections 2 and 3.
  • standard math Standard spectral and coercivity properties of the linearized Boltzmann operator L: self-adjointness, Fredholm alternative, coercivity on the null space complement.
    Used in the a priori estimates, e.g. (2.7) and (3.4), with reference to [4] and [8]; not reproved in the paper.
  • standard math Well-posedness of the auxiliary half-space problem with prescribed mass flux, Lemma 2.1.1 of [6].
    Invoked in Sections 1.2.1, 2, and 4.1 to construct solutions and to prove rotational symmetry of the layer.
  • ad hoc to paper The weighted L∞ boundary estimate of Lemma 3.3 in [15] transfers to the mixed Maxwell boundary when the diffuse part is treated as a source.
    This is the omitted proof of Lemma 2.1, which is load-bearing for the finite-slab estimates and hence for the d→∞ limit.

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Pith. "Pith review of Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits." pith.science (2026). https://pith.science/paper/FAJXEK6E

@misc{pith2026250108733,
  author       = {Pith},
  title        = {Pith review of: Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FAJXEK6E}},
  note         = {Machine review of arXiv:2501.08733}
}
abstract

This paper investigates the Knudsen layer equation in half-space, arising from the hydrodynamic limit of the Boltzmann equation to fluid dynamics. We consider the Maxwell reflection boundary condition with accommodation coefficient $0<\alpha<1$. We restrict our attention to hard sphere collisions with angular cutoff, proving the existence, uniqueness, and asymptotic behavior of the solution in $L^{\infty}_{x,v}$. Additionally, we demonstrate the application of our theorem to the hydrodynamic limit through a specific example. In this expample, we derive the boundary conditions of the fluid equations using our theorem and the symmetric properties of the Knudsen layer equation for $\alpha\in(0,1]$ and $\alpha=O(1)$. These derivations differs significantly from the cases of specular and almost specular reflection. This explicitly characterizes the {\em vanishing sources set} defined in \cite{jiang2024knudsenboundarylayerequations}

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

30 extracted references · 28 canonical work pages

  1. [18]

    Knudsen boundary layer equations for full ranges of cutoff collision kernels: Maxwell reflection boundary with all accommodation coefficients in [0,1]

    N. Jiang, Y.-L. Luo, and Y. Wu. Knudsen boundary layer eq uations for full ranges of cutoff collision kernels: Maxwell reflection boundary with all accommodatio n coefficients in [0,1]. arXiv:2407.02852

    work page Pith review arXiv

  2. [15]

    Huang, Z.-H

    F.-M. Huang, Z.-H. Jiang, and Y. Wang. Boundary layer so lution of the Boltzmann equation for specular boundary condition. Acta Math. Appl. Sin. Engl. Ser. , 39(1):65–94, 2023

    work page 2023

  3. [1]

    K. Aoki, C. Baranger, M. Hattori, S. Kosuge, G. Martal` o, J. Mathiaud, and L. Mieussens. Slip boundary conditions for the compressible Navier-Stokes equations. J. Stat. Phys. , 169(4):744–781, 2017

    work page 2017

  4. [2]

    Bardos, R

    C. Bardos, R. E. Caflisch, and B. Nicolaenko. The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Comm. Pure Appl. Math. , 39(3):323–352, 1986

    work page 1986

  5. [3]

    Bernhoff and F

    N. Bernhoff and F. Golse. On the boundary layer equations w ith phase transition in the kinetic theory of gases. Arch. Ration. Mech. Anal. , 240(1):51–98, 2021. 30 LING-BING HE, NING JIANG, AND YULONG WU

    work page 2021

  6. [4]

    R. E. Caflisch. The fluid dynamic limit of the nonlinear Bol tzmann equation. Comm. Pure Appl. Math. , 33(5):651–666, 1980

    work page 1980

  7. [5]

    Chen, T.-P

    C.-C. Chen, T.-P. Liu, and T. Yang. Existence of boundary layer solutions to the Boltzmann equation. Anal. Appl. (Singap.) , 2(4):337–363, 2004

    work page 2004

  8. [6]

    Coron, F

    F. Coron, F. Golse, and C. Sulem. A classification of well- posed kinetic layer problems. Comm. Pure Appl. Math., 41(4):409–435, 1988

    work page 1988

Show all 30 references
  1. [7]

    S. Deng, W. Wang, and S.-H. Yu. Pointwise convergence to K nudsen layers of the Boltzmann equation. Comm. Math. Phys. , 281(2):287–347, 2008

    2008

  2. [8]

    Desvillettes and F

    L. Desvillettes and F. Golse. A remark concerning the Cha pman-Enskog asymptotics. In Advances in kinetic theory and computing , volume 22 of Ser. Adv. Math. Appl. Sci. , pages 191–203. World Sci. Publ., River Edge, NJ, 1994

    1994

  3. [9]

    Esposito, Y

    R. Esposito, Y. Guo, C. Kim, and R. Marra. Non-isothermal boundary in the Boltzmann theory and Fourier law. Comm. Math. Phys. , 323(1):177–239, 2013

    2013

  4. [10]

    Golse, B

    F. Golse, B. Perthame, and C. Sulem. On a boundary layer p roblem for the nonlinear Boltzmann equation. Arch. Rational Mech. Anal. , 103(1):81–96, 1988

    1988

  5. [11]

    Golse and F

    F. Golse and F. Poupaud. Stationary solutions of the lin earized Boltzmann equation in a half-space. Math. Methods Appl. Sci. , 11(4):483–502, 1989

    1989

  6. [12]

    Y. Guo. Decay and continuity of the Boltzmann equation i n bounded domains. Arch. Ration. Mech. Anal. , 197(3):713–809, 2010

    2010

  7. [13]

    Hattori and S

    M. Hattori and S. Takata. Asymptotic theory for the time -dependent behavior of a slightly rarefied gas over a smooth solid boundary. J. Stat. Phys. , 147:1182–1215, 2012

    2012

  8. [14]

    Huang and Y

    F. Huang and Y. Wang. Boundary layer solution of the Bolt zmann equation for diffusive reflection bound- ary conditions in half-space. SIAM J. Math. Anal. , 54(3):3480–3534, 2022

    2022

  9. [16]

    Jiang, Y.-L

    N. Jiang, Y.-L. Luo, and S. Tang. Compressible Euler lim it from Boltzmann equation with Maxwell reflection boundary condition in half-space. arXiv:2101.11199

    arXiv

  10. [17]

    Jiang, Y.-L

    N. Jiang, Y.-L. Luo, and S. Tang. Compressible Euler lim it from Boltzmann equation with complete diffusive boundary condition in half-space. Trans. Amer. Math. Soc. , 377(8):5323–5359, 2024

    2024

  11. [19]

    Jiang and N

    N. Jiang and N. Masmoudi. Boundary layers and incompres sible Navier-Stokes-Fourier limit of the Boltz- mann equation in bounded domain I. Comm. Pure Appl. Math. , 70(1):90–171, 2017

    2017

  12. [20]

    Liu and S.-H

    T.-P. Liu and S.-H. Yu. Invariant manifolds for steady B oltzmann flows and applications. Arch. Ration. Mech. Anal., 209(3):869–997, 2013

    2013

  13. [21]

    Sakamoto, M

    S. Sakamoto, M. Suzuki, and K. Z. Zhang. Boundary layers of the Boltzmann equation in three-dimensional half-space. J. Differential Equations , 314:446–472, 2022

    2022

  14. [22]

    Sammartino and R

    M. Sammartino and R. E. Caflisch. Zero viscosity limit fo r analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equation s. Comm. Math. Phys. , 192(2):433–461, 1998

    1998

  15. [23]

    Sammartino and R

    M. Sammartino and R. E. Caflisch. Zero viscosity limit fo r analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solu tion. Comm. Math. Phys. , 192(2):463–491, 1998

    1998

  16. [24]

    Y. Sone. Kinetic theory and fluid dynamics . Modeling and Simulation in Science, Engineering and Tech- nology. Birkh¨ auser Boston, Inc., Boston, MA, 2002

    2002

  17. [25]

    Y. Sone. Molecular gas dynamics . Modeling and Simulation in Science, Engineering and Techn ology. Birkh¨ auser Boston, Inc., Boston, MA, 2007. Theory, techni ques, and applications

    2007

  18. [26]

    S. Ukai, T. Yang, and S.-H. Yu. Nonlinear boundary layer s of the Boltzmann equation. I. Existence. Comm. Math. Phys. , 236(3):373–393, 2003

    2003

  19. [27]

    S. Ukai, T. Yang, and S.-H. Yu. Nonlinear stability of bo undary layers of the Boltzmann equation. I. The case M∞ < − 1. Comm. Math. Phys. , 244(1):99–109, 2004

    2004

  20. [28]

    W. Wang, T. Yang, and X. Yang. Nonlinear stability of bou ndary layers of the Boltzmann equation for cutoff hard potentials. J. Math. Phys. , 47(8):083301, 15, 2006

    2006

  21. [29]

    W. Wang, T. Yang, and X. Yang. Existence of boundary laye rs to the Boltzmann equation with cutoff soft potentials. J. Math. Phys. , 48(7):073304, 21, 2007

    2007

  22. [30]

    X. Yang. The solutions for the boundary layer problem of Boltzmann equation in a half-space. J. Stat. Phys., 143(1):168–196, 2011. BOUNDARY LAYER EQUATION 31 (Ling-Bing He) Department of Mathematical Sciences, Tsinghua University , Beijing, 100084, P.R. China Email address : h...

    2011

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