{"id":"79583dab-dbed-431e-be59-ce2223ebff33","arxiv_id":"2501.08715","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives beta-dependent slip boundary conditions for compressible Navier-Stokes from the Boltzmann equation with almost specular Maxwell reflection, and rigorously proves the specular-reflection approximation in bounded domains.","lead":"This paper derives slip boundary conditions for the compressible Navier-Stokes equations from the Boltzmann equation with almost specular Maxwell reflection at walls, and it rigorously justifies the standard specular-reflection case in bounded domains. A smart generalist should care because this replaces imposed wall conditions with conditions derived from the underlying kinetic model, with explicit error estimates.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5 omits Chen-Kim's angular-momentum condition (4.7) for axisymmetric domains; without it the macroscopic Lemma 4.2 fails and the remainder bound collapses for admissible well-prepared data.","rationale":"The paper has two distinct claims: a formal derivation of slip boundary conditions for general accommodation coefficients (Theorem 1.1) and a rigorous CNS approximation for specular reflection (Theorem 1.5). The formal part is carefully executed: the exponent iota = 1 - beta emerges consistently from the solvability condition (2.43), the signs of the slip coefficients (2.45)-(2.46) are compatible with dissipativity, and the beta = 1 case is handled separately because the Taylor expansion in epsilon^iota degenerates. I do not see an internal inconsistency there. The rigorous part is another matter. The proof of Theorem 1.5 depends on Lemma 4.8, whose proof uses Lemma 4.2 (from Chen and Kim [15]) to control the macroscopic component (a,b,c). Lemma 4.2 is stated with hypothesis (4.7): for axisymmetric domains, the initial remainder must be orthogonal to all infinitesimal rotations. Lemma 4.8 carries this assumption. But Theorem 1.5 states 'any smooth bounded domain' and characterizes well-prepared initial data only by the ansatz (1.17); condition (4.7) is absent. The paper never proves that (1.17) implies (4.7), and it does not: with Omega the unit ball, (rho0,u0,theta0) = (1,0,1), and R_in = delta (x cross v) dot e sqrt(mu), the ansatz holds with small delta, yet the angular-momentum projection is positive, violating (4.7). Since this mode is conserved by the specular-reflection remainder equation, the space-time macroscopic estimate (4.8) cannot hold, and the bootstrap leading to (4.16) and (1.18)-(1.19) breaks down. Thus Theorem 1.5, as stated, is not established for axisymmetric domains. The fix is straightforward: include (4.7) in the theorem hypotheses or restrict the theorem to domains with dim R_Omega = 0. This does not invalidate the formal Theorem 1.1, and the analytical framework is otherwise coherent; hence a conditional acceptance remains appropriate.","tokens_in":53282,"tokens_out":23339,"duration_ms":209006,"concrete_test":"Verify whether (4.7) follows from the well-prepared ansatz (1.17). Concretely: set Omega = unit ball, (rho0,u0,theta0) = (1,0,1), G_in = 0, and R_in = delta (x cross v) dot e sqrt(mu) with delta small. This satisfies the smallness hypotheses of Lemma 4.8 but gives integral R_in v dot (Mx) sqrt(mu) dxdv = delta ||x cross e||^2_L2_x ||v_t||^2_L2_v > 0, so (4.7) fails. Run the proof of Lemma 4.2 for this datum and check whether the L2 macroscopic estimate (4.8) can hold despite the nonzero conserved angular momentum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The rigorous half (Theorem 1.5) is not proven for the full class of domains it claims. The macroscopic estimates in Lemma 4.2 are imported from Chen and Kim [15] and require the angular-momentum orthogonality condition (4.7) whenever the domain is axisymmetric (dim R_Omega != 0). Lemma 4.8, which constructs the remainder and yields (1.18), explicitly assumes (4.7) for such domains. However, Theorem 1.5 states 'any smooth bounded domain' and defines well-prepared initial data solely by (1.17); condition (4.7) is neither included in the theorem nor shown to follow from the well-prepared construction. For the unit ball with equilibrium CNS data (rho0,u0,theta0)=(1,0,1), one can take R_in = delta (x cross v) dot e sqrt(mu) with delta small: the smallness hypotheses of Lemma 4.8 hold, but the angular-momentum projection is positive, so (4.7) fails. The conserved angular momentum then prevents the L2 macroscopic estimate (4.8) from holding, so the remainder bound (4.16) and hence (1.18)-(1.19) collapse for this admissible initial datum. The fix is to add (4.7) to the hypotheses of Theorem 1.5 or restrict the theorem to domains with dim R_Omega = 0.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper has two main parts. In the formal part, the authors derive slip boundary conditions for the compressible Navier-Stokes-Fourier system from the scaled Boltzmann equation with Maxwell reflection boundary when the accommodation coefficient is α_ε = 0 or α_ε = χ ε^β, β > 0. They propose an ansatz with u - u_w = O(ε^ι), θ - θ_w = O(ε^ι), determine ι = 1 - β from the solvability of the Knudsen layer equation, and obtain the complete slip conditions (1.8) for β > 1, the slip conditions (1.9) for 0 < β < 1, and the critical condition (1.10) for β = 1, with coefficients (2.45) and (2.46). In the rigorous part, the paper proves a compressible Navier-Stokes approximation theorem for the Boltzmann equation with specular reflection: under smallness of the CNS data of order O(ε^{3/2}) and well-prepared Boltzmann data, the difference between the Boltzmann solution and the Chapman-Enskog expansion is O(ε^2) in L^2_{x,v} and O(ε) in a weighted L^∞_{x,v} norm. The rigorous proof combines conormal energy estimates for the CNS with the L^2-L^6-L^∞ framework for the remainder equation, using macroscopic estimates imported from Chen and Kim [15].","tokens_in":53623,"tokens_out":7448,"duration_ms":76481,"significance":"If the results are correct, this is a useful contribution: it completes the formal program of Aoki et al. [2] for almost specular Maxwell reflection, gives explicit formulas for the slip coefficients, and provides the first rigorous CNS approximation theorem with a boundary condition derived from the Boltzmann equation in the specular case. The paper makes good use of existing technology — conormal Sobolev spaces, Helmholtz decomposition, Agmon-Douglis-Nirenberg elliptic estimates, and the L^2-L^6-L^∞ framework — and it states its main theorems clearly. The formal derivation has a genuine new ingredient in the ansatz for general β > 0, and the rigorous part gives a self-contained energy structure modulo the imported macroscopic lemma. The main caveat is that the rigorous Theorem 1.5 is stated more broadly than Lemma 4.8, which is the actual estimate used in the proof; this is a fixable but load-bearing gap.","major_comments":[{"comment":"Theorem 1.5 is not proved for the full class of domains it states. Lemma 4.8, which constructs the remainder R and yields (1.18), assumes condition (4.7) whenever the domain is axisymmetric, i.e. dim R_Ω ≠ 0. Theorem 1.5, however, states 'any smooth bounded domain' and defines well-prepared initial data solely by (1.17) and the smallness condition on (ρ0, u0, θ0); it neither includes (4.7) nor shows that (1.17) implies it. The skeptic's counterexample is valid: for the unit ball, R_in = δ (x × v) · e √μ with small δ satisfies the smallness hypotheses of Lemma 4.8 but violates (4.7), since its angular-momentum projection is nonzero. The conserved angular momentum then prevents the macroscopic estimate (4.8) from holding, so the bound (1.18) and the approximation (1.19) collapse for this admissible datum. The fix is to add (4.7) to the hypotheses of Theorem 1.5 or to restrict the theorem to domains with dim R_Ω = 0.","section":"§1.4.2 and §4.3 (Theorem 1.5 vs. Lemma 4.8)"},{"comment":"The claimed β-continuity of the boundary conditions is not supported by the displayed formulas. For 0 < β < 1, the derivation of (1.9) uses u - u_w = O(ε^{1-β}) and θ - θ_w = O(ε^{1-β}), with coefficients b_I^u and b_I^θ evaluated at θ_w and with no quadratic velocity term. In the critical case β = 1, the boundary condition (1.10) contains the additional term |u - u_w|^2/4 and coefficients c_I^u and c_I^θ evaluated at θ_B rather than θ_w. Since ε^{1-β} → 1 as β → 1⁻ and u - u_w is O(1) in that limit, the quadratic term does not vanish, and b_I(θ_w) does not coincide with c_I(θ_B) unless θ_B = θ_w. Remark 1.3 should be revised, or the precise scaling under which (1.9) tends to (1.10) should be stated.","section":"§1.4.1, Remark 1.3 and §2.3"},{"comment":"Several estimates that carry the proof of Theorem 3.1 are delegated to 'tedious but trivial' or 'routine' computations. In Lemma 3.4 the bounds for the commutator terms involving C_i^α are asserted without details, and the proof of Lemma 3.6 states that the remaining terms are controlled by 'standard estimates' that are omitted. Since (3.13) and (3.37) are the base estimates for Corollary 3.8 and hence for the uniform regularity used in Theorem 1.5, these steps should be written out or replaced by a precise reference to the corresponding estimates in [18] or [47]. Without this, the rigorous part is not fully verifiable as written.","section":"§3.1-§3.2 (proofs of Lemmas 3.4 and 3.6)"}],"minor_comments":[{"comment":"The far-field condition in (2.9) is written as 'F^bb → 0 as η → 0'; the intended statement is η → ∞.","section":"§2.2.1, equation (2.9)"},{"comment":"In the completing-proof paragraph, '(1.18) directly follows from Lemma 4.16' should refer to Lemma 4.8, not Lemma 4.16.","section":"§4.3, proof of Theorem 1.5"},{"comment":"The notation 'L^2-L^6-L^8' is used in the abstract and in Section 4; the weighted bound in Lemma 4.6 is an L^∞ estimate, so the notation should be 'L^2-L^6-L^∞' or the symbol 8 should be explained as ∞.","section":"Abstract and §4"},{"comment":"Remark 1.4 says the Knudsen layer 'is not needed' for α_ε = 0 or β > 1, while Section 2.4 says the Knudsen layer appears at higher order; the wording should be clarified to 'not needed at first order'.","section":"§1.4.1 and §2.4"},{"comment":"The proof of Lemma 4.2 relies on [15] and includes a normalization argument with ̄a_0 and ̄c_0; the sentence 'Define ̄a_0 = ...' should explicitly state the conservation laws for mass and energy that justify the reduction, since these are not written out.","section":"§4.1, Lemma 4.2"}],"recommendation":"major_revision","confidential_remarks":"The gap in Theorem 1.5 is load-bearing but local: adding condition (4.7) or restricting the domain class fixes the rigorous result. The formal half is a solid contribution, and the paper deserves a chance after revision. The major comments above should be addressed before publication; the omitted routine estimates in Section 3 are a verification concern but not a reason to reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Formal half is genuinely new and the rigorous half is close, but Theorem 1.5 as stated is not true for all smooth bounded domains. The paper derives slip boundary conditions for the compressible Navier-Stokes system from the Boltzmann equation with Maxwell reflection when α_ε = 0 or α_ε = χ ε^β, and rigorously justifies the CNS approximation for the specular case. The formal part is the real contribution: the ansatz u−u_w = ε^{1−β} is new, the ε^{1−β} scaling in the slip laws is new, and the |u−u_w|^2 term at β = 1 completes the program started by Aoki et al. The order-balancing argument is coherent, and the consistency limits in Remark 1.3 are plausible, though the β→1 limit is asserted rather than demonstrated.\n\nThe rigorous half has a load-bearing gap. Lemma 4.2, imported from Chen and Kim, requires the angular-momentum orthogonality condition (4.7) whenever the domain is axisymmetric (dim R_Ω ≠ 0). Lemma 4.8, which constructs the remainder and yields (1.18), explicitly assumes (4.7) in that case. But Theorem 1.5 claims 'any smooth bounded domain' and defines well-prepared data only through (1.17). The condition (4.7) is neither stated in the theorem nor shown to follow from the well-prepared construction. The stress-test example is valid: in the unit ball with equilibrium CNS data, take R_in = δ (x×v)·e √μ with δ small. The hypotheses of Lemma 4.8 hold, but the angular-momentum projection is non-zero, so (4.7) fails, and the macroscopic estimate (4.8) cannot hold. The remainder bound (1.18), and hence the L^2 and L∞ convergences in (1.19), collapse for that admissible initial datum. The fix is simple: add (4.7) to the hypotheses of Theorem 1.5, or restrict the theorem to domains with dim R_Ω = 0.\n\nLesser issues: several 'routine' estimates in Section 3 are omitted, and the external dependence on [15] and [22] is honest and not circular. The paper does not itself prove the Knudsen-layer solvability, but that's a reasonable delegation for a derivation paper.\n\nFor whom: kinetic theory and fluid-limit people. It deserves a serious referee; the formal result is new enough to warrant publication even if the rigorous part needs revision. I would accept it for peer review and ask the authors to fix the domain assumption in Theorem 1.5.","headline":"Formal slip-boundary derivation is new and worth serious attention; the rigorous CNS justification overreaches by omitting the angular-momentum condition (4.7) needed for axisymmetric domains.","tokens_in":54071,"tokens_out":3360,"would_cite":true,"duration_ms":31703,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q20","76P05","76N06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives slip boundary conditions for the compressible Navier–Stokes–Fourier system from the Boltzmann equation under specular and almost-specular Maxwell reflection, and proves the specular approximation is accurate to…","keywords":["Compressible Navier-Stokes approximation","Chapman-Enskog expansion","Knudsen layer","Maxwell reflection boundary condition","Slip boundary conditions","Specular reflection","Conormal derivatives","Boltzmann equation"],"falsifier":"The decisive check is to evaluate the integrals in (2.43) with the source term (2.41) and confirm that they force $\\iota=1-\\beta$ and negative coefficients, since any other outcome would make the derived slip scaling (1.9) false.","tokens_in":53095,"feed_emoji":"🌬️","tokens_out":15275,"duration_ms":136908,"temperature":0.7,"pith_summary":"The paper’s goal is to close the gap between kinetic and fluid descriptions at walls: it derives the correct boundary conditions for the compressible Navier–Stokes–Fourier system from the Boltzmann equation when the wall reflects particles almost specularly, and it proves that the Navier–Stokes approximation is accurate in bounded domains when the reflection is exactly specular. The derivation splits into three regimes controlled by the accommodation coefficient $\\alpha_\\varepsilon=0$ or $\\alpha_\\varepsilon=\\chi\\varepsilon^\\beta$: $\\beta>1$ gives the classical complete-slip conditions, $0<\\beta<1$ gives slip of size $\\varepsilon^{1-\\beta}$ with coefficients fixed by a boundary-layer solvability condition, and $\\beta=1$ gives $\\varepsilon$-slip with an additional quadratic wall-slip temperature term. For specular reflection the paper proves an $O(\\varepsilon^2)$ $L^2$ error and $O(\\varepsilon)$ weighted $L^\\infty$ error between the Boltzmann solution and the Chapman–Enskog expansion, provided the Navier–Stokes data start $O(\\varepsilon^{3/2})$ close to equilibrium. The interest is that boundary conditions previously input by hand or left open are now consequences of the kinetic equation.","feed_headline":"Derived: slip boundary laws for every nearly specular wall","feed_subtitle":"Complete slip for β>1, ε^{1−β} slip for β<1, plus a rigorous O(ε²) bound for the specular case.","key_machinery":"The machinery is the first-order Chapman–Enskog ansatz $F_\\varepsilon=M+\\varepsilon G$, with $M$ the local Maxwellian and $G$ the Navier–Stokes correction, supplemented by a Knudsen layer correction $\\varepsilon F^{bb}$ when the wall accommodation is not negligible at order $\\varepsilon$. The layer correction is governed by the half-space kinetic equation $(\\xi\\cdot n)\\partial_y f^{bb}=L_{\\theta_w} f^{bb}$, and the solvability criterion (2.43) converts the requirement that the layer decays at infinity into explicit integrals that determine the slip coefficients $b^I_u,b^I_\\theta,c^I_u,c^I_\\theta$. For the rigorous part, the load-bearing device is the decomposition of the remainder $R$ into macroscopic part $PR$ and microscopic part $(I-P)R$, with a macroscopic $L^2$–$L^6$ estimate imported as Lemma 4.2, an $L^\\infty$ estimate along specular backward characteristics, and conormal-energy estimates for the Navier–Stokes system based on Helmholtz decomposition and elliptic regularity for the Stokes-type system.","core_discovery":"On the formal side, Theorem 1.1 states that with Maxwell reflection and $\\alpha_\\varepsilon=0$ or $\\alpha_\\varepsilon=\\chi\\varepsilon^\\beta$, the first-order Chapman–Enskog expansion $F_\\varepsilon=M+\\varepsilon G$ forces: complete slip (1.8) when $\\beta>1$; slip velocity and temperature jumps of size $\\varepsilon^{1-\\beta}$ (1.9) with negative coefficients $b^I_u,b^I_\\theta$ given by (2.45) when $0<\\beta<1$; and $\\varepsilon$-slip conditions (1.10) with an extra $|u-u_w|^2/4$ term when $\\beta=1$. The transition is that for $\\beta>1$ the diffuse part of the wall reflection is too weak to matter at order $\\varepsilon$, so no Knudsen layer is needed, while for $0<\\beta\\le 1$ the layer must be solved and the solvability condition (2.43) selects the boundary data. On the rigorous side, Theorem 1.5 shows that for hard spheres with specular reflection in any smooth bounded domain, if the CNS solution starts within $O(\\varepsilon^{3/2})$ of $(1,0,1)$ and the Boltzmann data are well prepared, the remainder satisfies the uniform bound (1.18), giving an $L^2$ error of $O(\\varepsilon^2)$ and a weighted $L^\\infty$ error of $O(\\varepsilon)$.","pith_inferences":["The authors do not draw it out, but their three-regime result predicts an experimental signature: in a channel with almost specular walls, slip should first appear at scale $\\varepsilon^{1-\\beta}$ for $\\beta<1$ and should be undetectable at Navier–Stokes order for $\\beta>1$.","The same solvability-condition machinery should transfer to other wall mechanisms, such as incoming or temperature-dependent reflection, where the paper notes the derivation is open.","Because the $\\beta>1$ case needs no Knudsen layer, the rigorous remainder argument for the specular case is the natural template for a proof at $\\beta>1$, provided the Navier–Stokes regularity theorem adapts to the same complete-slip data.","The explicit integral formulas (2.45)–(2.46) let one test the theory quantitatively against direct-simulation Monte Carlo data for hard spheres without fitting any parameter."],"forward_implications":["For exactly specular or extremely weak accommodation ($\\beta>1$), the Navier–Stokes boundary data are fixed, not fitted: the wall sees complete slip (1.8).","For $0<\\beta<1$, the slip velocity and temperature jump at the wall are $O(\\varepsilon^{1-\\beta})$ and are set by the half-space layer integrals (2.45), so different collision kernels give different slip coefficients instead of free parameters.","At the critical $\\beta=1$, the wall temperature jump acquires a quadratic $|u-u_w|^2/4$ term that is invisible when $\\beta<1$ because the slip velocity is higher order.","For specular reflection, the approximation error in bounded domains is $O(\\varepsilon^2)$ in $L^2$ and $O(\\varepsilon)$ in weighted $L^\\infty$, matching the remainder scaling $\\varepsilon^2\\sqrt{\\mu}R$ with no first-order boundary-layer singularity.","Because the derived slip coefficients are negative, the boundary contribution to the Navier–Stokes energy is dissipative, the property needed for well-posedness of the fluid system with these boundary conditions."],"supporting_citations":[{"why":"Derives the analogous slip boundary conditions for complete diffusion $\\alpha=O(1)$, providing the method and the baseline the present three-regime result extends.","marker":"[2]"},{"why":"Supplies the macroscopic $L^2$–$L^6$ estimates for the linearized Boltzmann equation with specular reflection in bounded domains, on which Lemma 4.2 and therefore Theorem 1.5 rest.","marker":"[15]"},{"why":"Proves the solvability condition (2.43) for the half-space Knudsen layer, the mechanism converting layer solvability into the slip coefficients (2.45)–(2.46).","marker":"[22]"},{"why":"Sets up the bounded-domain CNS approximation for the Boltzmann equation with a given Dirichlet boundary condition; the present paper adapts its energy framework to the derived complete-slip boundary condition.","marker":"[18]"},{"why":"Provides the $L^\\infty$ estimate along specular backward characteristics and the bilinear estimates used in Lemmas 4.6–4.8 to control the remainder nonlinearly.","marker":"[28]"},{"why":"Establishes uniform conormal regularity for the full compressible Navier–Stokes system with Navier-type slip, used as Theorem 3.1.","marker":"[57]"},{"why":"Supplies the elliptic estimates, with the complementing condition verified in Appendix B, needed for the higher-order Navier–Stokes regularity lemmas.","marker":"[1]"}],"fun_headline_variants":["Slip boundary laws derived for all β, rigorously for specular","From Boltzmann to Navier-Stokes: slip for every β>0","β>1 complete slip, β<1 ε^{1-β} slip: derived","First general-β slip ansatz, plus O(ε²) rigorous bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that two imported results hold exactly as used: the existence of decaying solutions to the thin kinetic boundary-layer problem under the stated solvability criterion, and the $L^2$–$L^6$ control of the fluid-like part of the linearized Boltzmann remainder in bounded domains with specular reflection, including its normalization for axially symmetric domains; if either imported result fails, the theorem built on it collapses.","fun_headline_variants_meta":{"raw":{"variants":["Slip boundary laws derived for all β, rigorously for specular","From Boltzmann to Navier-Stokes: slip for every β>0","β>1 complete slip, β<1 ε^{1-β} slip: derived","First general-β slip ansatz, plus O(ε²) rigorous bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000668,"raw_usage":{"total_tokens":3125,"prompt_tokens":1100,"completion_tokens":2025,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":716,"completion_tokens_details":{"reasoning_tokens":1941}},"tokens_in":716,"tokens_out":2025,"duration_ms":14985,"temperature":1.0,"reasoning_tokens":1941,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:19:51.234690+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive check is to evaluate the integrals in (2.43) with the source term (2.41) and confirm that they force $\\iota=1-\\beta$ and negative coefficients, since any other outcome would make the derived slip scaling (1.9) false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the analogous slip boundary conditions for complete diffusion $\\alpha=O(1)$, providing the method and the baseline the present three-regime result extends."},{"cited_title":"Chen and C","cited_arxiv_id":null,"evidence_quote":"Supplies the macroscopic $L^2$–$L^6$ estimates for the linearized Boltzmann equation with specular reflection in bounded domains, on which Lemma 4.2 and therefore Theorem 1.5 rest."},{"cited_title":"Golse, B","cited_arxiv_id":null,"evidence_quote":"Proves the solvability condition (2.43) for the half-space Knudsen layer, the mechanism converting layer solvability into the slip coefficients (2.45)–(2.46)."},{"cited_title":"Duan and S","cited_arxiv_id":null,"evidence_quote":"Sets up the bounded-domain CNS approximation for the Boltzmann equation with a given Dirichlet boundary condition; the present paper adapts its energy framework to the derived complete-slip boundary condition."},{"cited_title":"Guo, Decay and continuity of the Boltzmann equation in bounded domains , Arch","cited_arxiv_id":null,"evidence_quote":"Provides the $L^\\infty$ estimate along specular backward characteristics and the bilinear estimates used in Lemmas 4.6–4.8 to control the remainder nonlinearly."},{"cited_title":"Wang, Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain , Arch","cited_arxiv_id":null,"evidence_quote":"Establishes uniform conormal regularity for the full compressible Navier–Stokes system with Navier-type slip, used as Theorem 3.1."},{"cited_title":"Agmon, A","cited_arxiv_id":null,"evidence_quote":"Supplies the elliptic estimates, with the complementing condition verified in Appendix B, needed for the higher-order Navier–Stokes regularity lemmas."}],"review_version":1}