{"id":"5f363759-172c-4913-bf84-8cfff65e7865","arxiv_id":"2501.08707","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The acoustic limit from the hard-sphere Boltzmann equation with Maxwell reflection boundary condition is proved for classical solutions for all accommodation coefficients 0<α≤1, with convergence rate ε^{1/4}.","lead":"This paper proves that, as the mean free path shrinks to zero, solutions of the Boltzmann equation with a wall that both reflects and diffusely scatters gas molecules converge to the acoustic (sound) equations, for every accommodation coefficient between 0 and 1. It is the first such proof for classical smooth solutions in this physically standard diffuse-reflection regime, and it rigorously constructs the fluid and boundary-layer corrections predicted by Sone.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 rests on pointwise Knudsen-layer estimates from the unpublished [22] for 0<α<1, and on an omitted construction of f_bb_4..6; the expansion (2.45) is not fully defined or verified.","rationale":"The reader's CONDITIONAL verdict is well aligned with the evidence: the paper supplies a detailed formal boundary-layer analysis, a standard L2-L8 remainder framework, and symmetry identities that support the structure of the construction, so the result is plausible. But the decisive input for the advertised full range 0<α<1 is the pointwise Knudsen-layer lemma taken from an unpublished companion paper; no proof of that lemma appears here, and the paper's own statement of it contains internal typos that undermine confidence in its careful verification. This is a correctness risk, not a disagreement with consensus: if Lemma 3.3 is wrong, incomplete, or has constants that degenerate as α varies, the construction of f_bb_k and the final theorem fail. A secondary but equally concrete gap is that the higher-order layer terms f_bb_4, f_bb_5, f_bb_6 are never explicitly constructed, so even granting Lemma 3.3, the expansion (2.45) is incomplete as written. The reader's weakest_assumption identified Lemma 3.3 as the key external input; I agree with that identification in substance, while additionally emphasizing the paper-internal omission of the k≥4 layer construction and the omitted proof of Proposition 3.5. The appropriate verdict remains CONDITIONAL: the conditions should be a published proof of Lemma 3.3 or a complete proof of Proposition 3.5 including the k≥4 construction, plus correction of the typographical errors in the statement of the Knudsen-layer problem.","tokens_in":31048,"tokens_out":12554,"duration_ms":122569,"concrete_test":"Obtain [22] and independently re-derive Lemma 3.3 for all α∈(0,1), tracking the dependence of the exponential rate σ0 and the constant C on α; verify that the estimates hold uniformly on α∈[α0,1] for every fixed α0>0 and that the only solvability condition is ∫_{v3<0} v3 g√μ dv=0. In the same pass, write out explicit formulas for f_bb_4, f_bb_5, f_bb_6 using the symmetry identities (B.1)-(B.5) and check that the resulting source terms and boundary data satisfy the hypotheses of Lemma 3.3; if any of these steps fails, the expansion (2.45) and Theorem 1.1 are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"To prove Theorem 1.1, every term in the truncation (2.45) must be constructed and uniformly bounded so that the source S in (2.46) is controlled. Proposition 3.5 asserts these bounds, but its proof is omitted, with the text sending the reader to [18, Proposition 5.1]. More specifically, §2.3 constructs the Knudsen-layer corrections f_bb_2 and f_bb_3 in detail, then states that f_bb_k (k≥4) 'can be constructed by the same ways' and that 'the calculations are tedious but trivial, so we will omit the details here.' Thus f_bb_4, f_bb_5, f_bb_6 appear in (2.45) and in S5 of (2.49) without being defined, so the remainder equation (2.46) is not a well-defined closed equation as written. The pointwise exponential decay estimates needed to solve the Knudsen-layer problems (2.36) for 0<α<1 are delegated to Lemma 3.3, quoted from the unpublished manuscript [22]; the published [23] covers only α=1. Lemma 3.3 also contains apparent typos, including 'lim_{ξ→0} f=0' in (3.4) and 'for all λ∈(0,λ0)' with no λ appearing in the statement, suggesting the imported lemma has not been carefully checked in this paper. The central claim, whose advertised novelty is precisely the full range 0<α<1, is therefore not self-contained at its most load-bearing point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an acoustic limit from the scaled Boltzmann equation with hard-sphere collisions and Maxwell reflection boundary condition in the half-space, for accommodation coefficient α in the full range 0<α≤1. The proof is based on a Hilbert expansion with viscous and Knudsen boundary layers, truncated at order ε^3, leaving a remainder f_R,ε whose L2 and weighted L∞ norms are controlled uniformly in ε. The main advertised novelty is the treatment of α=O(1), for which the Knudsen-layer mechanism differs from the previously studied cases α=0 and α=o(1). The paper states the main theorem (Theorem 1.1) and derives boundary conditions for acoustic system (2.31) and for the viscous layer, with slip coefficients b1,c1,b2,c2 determined by Milne-type problems (2.43)-(2.44). The final convergence rate is ε^{1/4} in the L2 norm and ε^{3/4} in the weighted L∞ norm (Remark 1.2). The overall structure is standard, but several load-bearing steps are asserted without proof and, in one case, delegated to an unpublished manuscript.","tokens_in":31320,"tokens_out":3371,"duration_ms":33073,"significance":"If the missing steps were supplied, the result would be significant: it would give the first rigorous classical-solution justification of the acoustic limit for the Boltzmann equation with Maxwell reflection boundary condition with accommodation coefficient of order one, thereby confirming Sone's formal analysis and extending the renormalized-solution result of Jiang-Levermore-Masmoudi to a setting where boundary layers are visible. The paper correctly identifies that the novel difficulty is the full range 0<α≤1 and the associated Knudsen-layer solvability and pointwise decay. However, the central technical input for this range is exactly what is not proved here. The paper also contains a useful, clearly written formal derivation of the boundary-layer hierarchy and the slip boundary conditions. The L2-L∞ remainder framework follows the established pattern of Guo-Huang-Wang and Guo-Jang-Jiang, and the estimates in Sections 4.1-4.2 are plausible, though they depend on the uniform bounds asserted in Proposition 3.5.","major_comments":[{"comment":"The terms f_bb_4, f_bb_5 and f_bb_6 appear in the expansion (2.45) and in the source term S5 in (2.49), but they are never defined. The text states in §2.3 that f_bb_k for k≥4 'can be constructed by the same ways' and that 'the calculations are tedious but trivial, so we will omit the details here.' As written, the remainder equation (2.46) is not a closed, well-defined equation, because S5 contains f_bb_5 and f_bb_6. This is a load-bearing gap: the L2 estimate in Lemma 4.1 uses the uniform bounds of these terms via Proposition 3.5, and the proof of Theorem 1.1 cannot proceed without a complete construction of the truncated expansion.","section":"§2.3, Eq. (2.45), Eq. (2.49)"},{"comment":"Lemma 3.3 is quoted from the unpublished manuscript [22] and is the only source of the pointwise exponential Knudsen-layer estimates for 0<α<1. The published reference [23] covers only α=1. Since the advertised novelty of the paper is precisely the full range 0<α≤1, this lemma is the central technical input of the paper. The lemma statement also contains apparent typos: the far-field condition in (3.4) is written as 'lim_{ξ→0} f=0' instead of ξ→∞, and the phrase 'for all λ∈(0,λ0)' appears with no λ in the statement. The paper should either include a proof of Lemma 3.3, or state it as an explicit assumption; as it stands, Theorem 1.1 rests on an unverifiable citation.","section":"Lemma 3.3, p. 17-18"},{"comment":"Proposition 3.5 asserts uniform Sobolev and weighted-L∞ bounds for all expansion terms f_k, f_b_k and f_bb_k for k=1,...,6, with a specific hierarchy of Sobolev indices and weights. The proof is omitted, with a reference to [18, Proposition 5.1]. This proposition is load-bearing: it is used to bound the source terms S3, S4 and S5 in the remainder estimate (4.1), and it must cover the new Knudsen-layer estimates for 0<α<1 from Lemma 3.3 and the new coupled boundary conditions derived in §2.3. The adaptation from [18] is not immediate, and the assertion that the bounds hold with the stated index hierarchy needs a detailed proof.","section":"Proposition 3.5, p. 19"},{"comment":"The proof of Lemma 3.2 for the heat-layer estimates is omitted, with the comment that it is 'similar and much easier as [18, Lemma 4.1].' Since the heat layer is a standard component and the lemma statement is specific, this omission is less serious than the previous ones, but the proof should still be included or a precise reference supplied, because the weighted Sobolev norms H^r_l defined in §1.3 are nonstandard and the compatibility conditions are not spelled out.","section":"Lemma 3.2, p. 17"}],"minor_comments":[{"comment":"In the definition of the remainder source term, 'S := S1 + S2 + S3 + S3 + S4 + S5' lists S3 twice; presumably the second S3 should be S3, but the duplicate suggests a typographical error that should be corrected.","section":"Eq. (2.49)"},{"comment":"The notation 'fR,ǫ' appears in the boundary term in (4.1) and in (4.3), inconsistent with the subscript 'fR,ε' used elsewhere; this should be corrected.","section":"Lemma 4.1, Eq. (4.1) and Eq. (4.3)"},{"comment":"The far-field boundary condition in the Knudsen-layer problem (3.4) is written as 'lim_{ξ→0} f pt, ¯x, ξ, v q = 0', while all other far-field conditions in the paper use ξ→∞; this is almost certainly a typo and should be fixed.","section":"Lemma 3.3, Eq. (3.4)"},{"comment":"The sentence 'The calculations are teidous but trival' contains spelling mistakes ('teidous' and 'trival') and should be rewritten as 'tedious but trivial'.","section":"§2.3, p. 15"},{"comment":"The velocity weight w_l is defined by w_l(v) = {1+|v|^2}^{l/2}, but in Lemma 4.3 the kernel estimate uses e^{η|v-v'|^2} and the weight w_l(v)/w_l(v'). The relationship between the polynomial weight and the exponential factor should be clarified at first use.","section":"§1.3, Eq. (1.11)-(1.12)"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem depends essentially on Lemma 3.3, which is quoted from the authors' own unpublished manuscript [22] (He, Jiang, Wu). Given that the same authors are involved, the citation pattern is understandable, but the referee report must emphasize that the present manuscript is not self-contained at the point of its advertised novelty. If [22] is not yet publicly available, the editor may wish to ask the authors to include the proof of Lemma 3.3 or to restrict the main theorem to the case α=1, for which the published reference [23] exists. The paper has a coherent formal structure and the remainder estimates are plausible, so a major revision addressing the missing definitions and proofs is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is the first serious attempt at the acoustic limit for α=O(1) in the classical-solution framework, and the mechanism is genuinely different from the α=0 and α=O(√ε) cases. It deserves a referee, but the proof as written is conditional on material that is not in the paper.\n\nWhat's new: previous rigorous results only handled specular or almost-specular reflection; here the Knudsen layer has one solvability condition rather than four, and the boundary conditions for the fluid and viscous layer come from the vanishing-at-infinity requirement. The formal Hilbert/viscous/Knudsen expansion is worked out in detail, and the L2-L8 remainder estimates are standard and largely present. The slip coefficients b1,c1,b2,c2 are determined by solving Milne/Knudsen problems, not by matching the acoustic limit, so there's no circularity.\n\nThe soft spots are real and load-bearing. Proposition 3.5, which is the uniform bound on all expansion terms, is asserted without proof; the reader is pointed to [18, Prop 5.1] for the specular case, but the whole point here is that α=O(1) changes the boundary structure. Likewise f_bb_4 through f_bb_6 are declared \"tedious but trivial\" and never defined, so the truncated expansion (2.45) is not fully described and the source term S5 in (2.49) is not a well-defined object as written. Most importantly, the pointwise exponential Knudsen-layer estimates for 0<α<1 are imported from the unpublished He-Jiang-Wu manuscript [22]; the published [23] only covers α=1. Lemma 3.3 also has typos—the limit in (3.4) should be ξ→∞, and \"for all λ∈(0,λ0)\" has no λ in the statement—which makes me think the import wasn't carefully checked. None of these is fatal by itself; they are the kind of gaps that a serious revision can close. But as it stands, Theorem 1.1 is not self-contained and the proof is not complete.\n\nWho this is for: specialists in Boltzmann hydrodynamic limits. If [22] is solid and the induction in Prop 3.5 can be written out, this would be a substantial advance. I'd send it to peer review and ask for the omitted proofs or a published reference for the Knudsen-layer estimates. I would not accept it as is.\n\nRecommendation: serious referee, major revision.","headline":"The α=O(1) acoustic limit is a real new result, but the proof is conditional on unpublished Knudsen-layer estimates and omitted construction of high-order terms; worth refereeing, not accepting as is.","tokens_in":31942,"tokens_out":2839,"would_cite":true,"duration_ms":27521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q20","35B25","82C40","76P05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For hard-sphere Boltzmann gases in a half-space with Maxwell reflection at any accommodation coefficient $0<\\alpha\\leq 1$, the paper proves convergence to the acoustic system with slip boundary condition $u_{1,3}=0$, at rate…","keywords":["Boltzmann equation","acoustic limit","Maxwell reflection boundary condition","accommodation coefficient","Knudsen layer","viscous boundary layer","Hilbert expansion","hard sphere collisions"],"falsifier":"Solve the linear half-space Knudsen-layer problem (3.4) for a fixed $\\alpha$ in $(0,1)$, for example $\\alpha=1/2$, with a smooth compactly supported source $s$ and boundary data $g$ satisfying the solvability condition, and test whether the solution decays exponentially in $\\xi$ with rate $\\sigma_0>0$ and whether the slip coefficients $(b_1,b_2,c)$ are finite; a single $\\alpha$ with non-exponential decay or no finite slip coefficients would disprove the key input behind Theorem 1.1.","tokens_in":30744,"feed_emoji":"🌊","tokens_out":7311,"duration_ms":66103,"temperature":0.7,"pith_summary":"The paper proves that, for hard-sphere gases near a flat wall, the scaled Boltzmann equation with Maxwell reflection converges to the linear acoustic system when the accommodation coefficient $\\alpha$ lies anywhere in $(0,1]$. Previous classical-solution results handled only specular reflection ($\\alpha=0$) or almost specular reflection ($\\alpha=O(\\sqrt{\\varepsilon})$), and the boundary-layer mechanism here is genuinely different. The construction combines an interior Hilbert expansion with a viscous layer of thickness $\\sqrt{\\varepsilon}$ and a Knudsen layer of thickness $\\varepsilon$, and the Knudsen layer's decay at infinity supplies the boundary conditions that the acoustic system alone cannot provide. The result is a smooth-solution analogue of the renormalized acoustic-limit work [25], and the first rigorous justification of Sone's formal boundary-layer analysis for $\\alpha=O(1)$.","feed_headline":"Acoustic limit proven for Boltzmann gas at all wall accommodations","feed_subtitle":"A three-layer construction with Knudsen and viscous boundary layers yields an optimal εⁱ⁄⁴ convergence rate.","key_machinery":"The expansion is organized at three scales: interior terms $F_k$ in $x$, viscous-layer terms $F^b_k$ in $\\zeta=x_3/\\sqrt{\\varepsilon}$, and Knudsen-layer terms $F^{bb}_k$ in $\\xi=x_3/\\varepsilon$. The load-bearing object is the linear Knudsen-layer system (2.36)--(2.37), $v_3\\partial_\\xi f^{bb}_k + L f^{bb}_k = S$, with the Maxwell boundary operator $K$; its solutions are built from fundamental solutions $\\varphi^{(1)}_1$ and $\\varphi^{(0)}_1$ whose existence fixes slip coefficients $b_1,c_1$ and hence the boundary conditions for the fluid and viscous layers. Lemma 3.3 supplies pointwise exponential-in-$\\xi$ estimates for these layers for all $0<\\alpha<1$, and the remainder equation (2.46) is controlled by $L^2$ energy estimates plus weighted $L^\\infty$ estimates along backward characteristics.","core_discovery":"The central claim is Theorem 1.1: for $0<\\alpha\\leq 1$, hard-sphere collisions, and well-prepared initial data, the Boltzmann equation (1.1) with Maxwell reflection (1.2) has a unique solution on $[0,\\tau]$ of the expanded form (2.45), and the remainder satisfies $\\sup_t(\\|f_{R,\\varepsilon}\\|_2^2 + \\sqrt{\\varepsilon}^3 \\|h_{R,\\varepsilon}\\|_\\infty) \\leq C$. Consequently $F_\\varepsilon - \\mu - \\sqrt{\\varepsilon}F_1$ tends to zero in the stated norms at rate $\\varepsilon^{1/4}$, where $(\\rho_1,u_1,\\theta_1)$ solves the acoustic system (2.10) with the slip-type boundary condition $u_{1,3}=0$. The decisive mechanism is that, for $\\alpha=O(1)$, the Knudsen-layer problem has only one algebraic solvability condition, yet the requirement that the layer vanish at infinity imposes four boundary conditions; the extra conditions are absorbed by a viscous Prandtl-type layer, whose boundary condition turns out to be Dirichlet rather than Neumann (the $\\alpha=0$ case) or Robin (the $\\alpha=O(\\sqrt{\\varepsilon})$ case).","pith_inferences":["The same three-layer ansatz should extend to general cutoff collision kernels; the paper states this is standard from the techniques in [19] and [26], so a direct check is to rerun Proposition 3.5 with general kernels.","As $\\alpha$ crosses from $O(1)$ to $o(1)$, the Dirichlet boundary condition should deform into the Robin or Neumann conditions of earlier works; quantifying this transition would require treating the singular limit $\\alpha\\to 0$, which the paper excludes.","Because the leading viscous layer satisfies linear heat-type equations here, the corresponding Euler limit with $\\alpha=O(1)$ would need the nonlinear compressible Prandtl equations, whose well-posedness the paper identifies as open; the acoustic result is the test case that avoids that obstruction.","The $\\varepsilon^{1/4}$ rate is an $L^2$ artifact of the layer's thickness; a sharper $\\varepsilon^{1/2}$ statement would require weighted or pointwise norms that track the layer, a refinement not attempted here."],"forward_implications":["For every small $\\varepsilon$, a unique solution exists on the fixed time interval $[0,\\tau]$ and collapses onto $\\mu+\\sqrt{\\varepsilon}F_1$ as $\\varepsilon\\to 0$, so the acoustic system with $u_{1,3}=0$ is the effective macroscopic model.","The rate $\\varepsilon^{1/4}$ is natural: the $\\sqrt{\\varepsilon}$-thick viscous layer contributes $L^2$ norm of order $\\varepsilon^{1/4}$, so no faster $L^2$ rate is possible without changing norms.","The boundary condition for the viscous layer is Dirichlet, completing the picture: Neumann for $\\alpha=0$, Robin for $\\alpha=O(\\sqrt{\\varepsilon})$, and Dirichlet for $\\alpha=O(1)$.","At order $\\sqrt{\\varepsilon}^2$ and higher, Knudsen-layer corrections are needed and constructed with exponential decay, giving purely boundary-controlled slip effects.","This is the first classical-solution justification of Sone's formal acoustic boundary-layer analysis for $\\alpha=O(1)$ in a compressible fluid model with boundary."],"supporting_citations":[{"why":"Supplies Lemma 3.3, the pointwise exponential Knudsen-layer estimates for $0<\\alpha<1$ on which the expansion and remainder estimates depend.","marker":"[22]"},{"why":"Sone's book provides the formal Hilbert-expansion boundary-layer analysis whose acoustic boundary conditions the paper makes rigorous.","marker":"[34]"},{"why":"Sone's molecular gas dynamics book supplies the isotropic-function identities and formal Knudsen-layer structure used in Section 2.","marker":"[35]"},{"why":"The specular-reflection predecessor $\\alpha=0$; its Hilbert expansion, layer construction, and energy method are the templates adapted here.","marker":"[18]"},{"why":"Golse--Perthame--Sulem classification of kinetic layer problems; gives the four-solvability-conditions context and well-posedness framework.","marker":"[13]"},{"why":"The renormalized-solution acoustic limit with Maxwell reflection; the paper's smooth-solution analogue, and source of the $u_{1,3}=0$ condition.","marker":"[25]"},{"why":"Caflisch's Hilbert-expansion fluid limit provides the truncation and remainder strategy and the $L^2$ framework.","marker":"[8]"},{"why":"Acoustic limit without boundary; supplies the $L^2$--$L^\\infty$ bootstrap and backward-characteristic machinery used for the remainder.","marker":"[19]"},{"why":"Almost specular $\\alpha=O(\\sqrt{\\varepsilon})$ case; the Robin boundary condition that the $\\alpha=O(1)$ Dirichlet result contrasts with.","marker":"[27]"}],"fun_headline_variants":["Acoustic limit proven for every wall reflection strength","Boltzmann acoustic limit extends to full accommodation range","Knudsen-viscous layers yield acoustic limit for all alpha","Acoustic limit from Boltzmann with general Maxwell boundary"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction leans on Lemma 3.3, imported without proof from the unpublished manuscript [22]: pointwise exponential decay estimates for the Knudsen-layer problem for every $0<\\alpha<1$; if those estimates fail for some $\\alpha$ in that range, the boundary-layer expansion and the theorem collapse.","fun_headline_variants_meta":{"raw":{"variants":["Acoustic limit proven for every wall reflection strength","Boltzmann acoustic limit extends to full accommodation range","Knudsen-viscous layers yield acoustic limit for all alpha","Acoustic limit from Boltzmann with general Maxwell boundary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001082,"raw_usage":{"total_tokens":4542,"prompt_tokens":978,"completion_tokens":3564,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":3501}},"tokens_in":594,"tokens_out":3564,"duration_ms":27786,"temperature":1.0,"reasoning_tokens":3501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:19:27.147313+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the linear half-space Knudsen-layer problem (3.4) for a fixed $\\alpha$ in $(0,1)$, for example $\\alpha=1/2$, with a smooth compactly supported source $s$ and boundary data $g$ satisfying the solvability condition, and test whether the solution decays exponentially in $\\xi$ with rate $\\sigma_0>0$ and whether the slip coefficients $(b_1,b_2,c)$ are finite; a single $\\alpha$ with non-exponential decay or no finite slip coefficients would disprove the key input behind Theorem 1.1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.3, the pointwise exponential Knudsen-layer estimates for $0<\\alpha<1$ on which the expansion and remainder estimates depend."},{"cited_title":"Sone, Kinetic theory and fluid dynamics , Modeling and Simulation in Science, Engineering and Tech- nology, Birkh¨ auser Boston, Inc., Boston, MA, 2002","cited_arxiv_id":null,"evidence_quote":"Sone's book provides the formal Hilbert-expansion boundary-layer analysis whose acoustic boundary conditions the paper makes rigorous."},{"cited_title":"Sone, Molecular gas dynamics , Modeling and Simulation in Science, Engineering and Techn ology, Birkh¨ auser Boston, Inc., Boston, MA, 2007","cited_arxiv_id":null,"evidence_quote":"Sone's molecular gas dynamics book supplies the isotropic-function identities and formal Knudsen-layer structure used in Section 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The specular-reflection predecessor $\\alpha=0$; its Hilbert expansion, layer construction, and energy method are the templates adapted here."},{"cited_title":"Golse, B","cited_arxiv_id":null,"evidence_quote":"Golse--Perthame--Sulem classification of kinetic layer problems; gives the four-solvability-conditions context and well-posedness framework."},{"cited_title":"Jiang, C","cited_arxiv_id":null,"evidence_quote":"The renormalized-solution acoustic limit with Maxwell reflection; the paper's smooth-solution analogue, and source of the $u_{1,3}=0$ condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Caflisch's Hilbert-expansion fluid limit provides the truncation and remainder strategy and the $L^2$ framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Acoustic limit without boundary; supplies the $L^2$--$L^\\infty$ bootstrap and backward-characteristic machinery used for the remainder."}],"review_version":1}