{"id":"2bf56e33-14ea-4bda-83d5-5756e0b112af","arxiv_id":"2411.17068","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Global solutions near Maxwellians are constructed for the Boltzmann equation in an infinite layer with diffuse reflection boundaries, with heat-equation-type decay in the 3D case and existence without decay in the 2D case.","lead":"This paper proves global-in-time existence and polynomial decay for the Boltzmann equation in an infinite flat layer with diffusely reflecting, isothermal, stationary walls. It is the first such result for non-compact diffuse boundaries and matches the decay rate of the two-dimensional heat equation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stochastic-cycle survival estimates (Lemmas 9 and 17) are deferred to [27] without proof; since the L∞ bounds in Proposition 6 and Lemma 16 depend on them, the main theorems are conditional pending a self-contained derivation.","rationale":"I read the paper in good faith and checked the central proof structure. The claimed novelty—global solutions with heat-equation-type decay for the Boltzmann equation in an infinite layer with diffuse boundaries—is supported by a coherent combination of Fourier transform in the horizontal variables, L1_k ∩ Lp_k energy estimates, a dual test-function argument for macroscopic dissipation, and stochastic-cycle L∞ estimates. The macroscopic dissipation estimates and the time-decay interpolation are carried out in detail and appear internally consistent; the minor typographical slips in Lemma 8 do not affect the argument. The single most load-bearing unresolved point is the survival probability for stochastic cycles with many diffuse reflections: Lemmas 9 and 17 are not proved in the manuscript but are essential for the L∞ estimates that close the nonlinear iteration. The reader's weakest-assumption analysis identified exactly this point, and I agree that it is the right concern. Because the 1D slab geometry makes the estimate plausible and likely adapts from [27] with minor changes, the appropriate disposition is a conditional acceptance: the authors should supply the missing proof or an exact transfer argument from [27]. My verdict therefore remains unchanged from the reader's CONDITIONAL assessment; I do not see evidence of a deeper flaw that would require rejection or a new concern beyond the deferred stochastic-cycle proof.","tokens_in":83308,"tokens_out":29943,"duration_ms":253661,"concrete_test":"Provide a self-contained proof of Lemma 9 for the slab: let U_i be i.i.d. with density u e^{-u^2/2} on (0,∞); show that for n = C1 T0^{5/4}, P(Σ_{i=1}^{n} 2/U_i ≤ T0) ≤ (1/2)^{C2 T0^{5/4}}. This can be settled analytically via a large-deviation lower-tail estimate: survival requires all U_i ≳ T0^{1/4}, giving probability exp(−c T0^{7/4}), which is stronger than the claimed bound. Alternatively, run a Monte Carlo simulation for T0 = 20, 40, 80 with 10^6 samples of the 1D stochastic cycle; if the empirical survival probability ever exceeds (1/2)^{C2 T0^{5/4}} for the specified C2, the lemma fails and the L∞ argument in Proposition 6 and Lemma 16 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central L∞ control in both Theorem 1 and Theorem 3 is obtained via stochastic cycles: Proposition 6 and Lemma 16. The critical inputs are Lemma 9 and Lemma 17, which assert that after n = C1 T0^{5/4} diffuse reflections the probability that the backward cycle survives is at most (1/2)^{C2 T0^{5/4}}. In the manuscript, Lemma 9 is dismissed with 'The proof is similar to [27]' and Lemma 17 with 'the same as Lemma 9'; no proof is given. These estimates are genuinely load-bearing: Lemma 10 and Lemma 18 use them to make the multi-reflection boundary terms (4.61) and (6.13) a factor o(1), and without that absorption the L1_k L∞_{T,x3,v} estimate in Proposition 6 (and its analogue Lemma 16) does not close. Although the 1D slab geometry makes an adaptation of [27] plausible—crossing times depend only on v3—the survival bound is a quantitative statement about the diffuse-reflection kernel and needs to be re-derived in this setting; the constants and rates in [27] are for a bounded 3D domain and do not automatically transfer. A secondary gap is the 'standard sequential argument' for existence, summarized in one sentence in §4.4, but the stochastic-cycle lemma is the more specific obstruction. The remainder of the proof—the Fourier-side L1∩Lp interpolation, the dual test-function argument, and the nonlinear estimates in Lemma 3—is coherent and internally consistent on inspection, with only minor typographical slips in Lemma 8.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the nonlinear Boltzmann equation (hard-sphere collisions) with isothermal diffuse-reflection boundaries in the infinite layers R^2 x (-1,1) and R x (-1,1). Theorem 1 constructs global-in-time solutions near global Maxwellians for initial perturbations small in a weighted L^1_k cap L^p_k (Fourier-side) space and proves polynomial decay with exponent sigma/2, where sigma = 2(1-1/p)-2epsilon > 1, stated as being the same as the two-dimensional heat equation decay; Theorem 2 refines the low-frequency macroscopic dissipation estimates for the components of the fluid part using time-derivative estimates; Theorem 3 gives global existence in the two-dimensional layer via an L^2 cap L^infty argument in physical space, without a decay rate. The strategy combines a horizontal Fourier transform, the L^1_k cap L^p_k method of [15], a frequency-weighted test-function argument for the macroscopic dissipation, and Guo's L^2-L^infty boundary framework [27] with stochastic cycles for the pointwise-in-velocity control.","tokens_in":83524,"tokens_out":15497,"duration_ms":136847,"significance":"If the deferred stochastic-cycle estimates hold, this is a substantial result: the first global-in-time solutions with heat-equation-type decay for the Boltzmann equation with non-compact diffuse boundaries, extending the L^1 cap L^p Fourier method to an initial-boundary-value problem and connecting with Kagei's layer results for the compressible Navier-Stokes equations. The Fourier-side macroscopic estimates (Lemmas 5-8), the time-weighted energy estimates, and the physical-space L^2-L^infty argument for Theorem 3 are worked out in long-form detail and appear internally coherent; the construction of test functions with the frequency weight |k|^2/(1+|k|^2) and the Poincare-based refinements for b1, b2 are genuinely non-trivial. The paper also deserves credit for being explicit about the limitations: no decay in Theorem 3, an epsilon-loss in the rate, and the low-frequency degeneracy issues flagged in Remarks 1-5. However, the two load-bearing probability lemmas (Lemmas 9 and 17) are not proved in the manuscript, so the significance is conditional on their validity.","major_comments":[{"comment":"The stress-test concern lands: Lemmas 9 and 17 are load-bearing and their proofs are deferred. Lemma 10 uses Lemma 9 to make the multi-reflection boundary term (4.61) of order o(1), and Lemma 18 uses Lemma 17 for the analogous term (6.13); without those absorptions the weighted L^1_k L^infty_{T,x3,v} estimate in Proposition 6 (and its analogue Lemma 16) does not close, and therefore Theorems 1-3 are conditional. The statements assert a quantitative survival bound for n = C_1 T_0^{5/4} reflections with probability at most (1/2)^{C_2 T_0^{5/4}}, but the proofs are replaced by 'The proof is similar to [27]' and 'the same as Lemma 9, since the backward exit time tb(x,v) in both settings are determined by v3'. The geometric reduction is plausible because the crossing times depend only on x3 and v3, but it is not automatic: reference [27] is a bounded three-dimensional domain, and the constants, the exponent 5/4, and the estimates on the diffuse-reflection kernel must be re-derived or mapped precisely to the slab with the same measure dsigma = sqrt(2pi) mu |v3| dv and with constants uniform in the horizontal variables. Please include the full derivation or a detailed reduction to [27] with all constants tracked; as written, this is an external load-bearing assumption rather than a proved estimate.","section":"Section 4.3, Lemma 9; Section 6.2, Lemma 17"},{"comment":"The existence step is summarized in a single sentence ('standard sequential argument') with positivity referred to [15] and [18]. Since the a priori estimates of Propositions 7 and 9 contain quadratic terms that are absorbed only by the smallness of the initial data, and since the norms in (1.11)-(1.12) involve L^1_k and L^infty_{T,x3,v} quantities that must pass to the limit along an approximating sequence (e.g., velocity cutoffs or regularized boundary data), a more detailed outline of the approximation, the uniform-in-sequence bounds, and the limit passage would make the global-existence claim verifiable from the manuscript itself. This is not presented as evidence of a flaw, but the central claim is existence, and the current level of detail is thinner than the rest of the paper.","section":"Sections 4.4 and 6.3"}],"minor_comments":[{"comment":"In the proof of (4.13), equations (4.39) and (4.40) bound expressions involving b3 on the left but write on the right 'o(1) ||b1||^2'; similarly, the closing paragraph of the proof of (4.14) says the extra term is controlled 'by the same computation in (4.11)' while writing psi_b and a norm of b3 where psi_c and c are clearly intended. These are local typos but they make the refined estimates hard to follow.","section":"Section 4.2, Lemma 8"},{"comment":"The weight |k|/sqrt(1+|k|^2) is consistently typeset as '|k| p 1+|k|2' in displayed equations, which obscures the estimates; the same applies to the stray superscript 2 in '|(I-P_gamma)f|^2' on the L^1_k L^2_{T,gamma+} terms in Lemma 4 and elsewhere, where the square is a typo.","section":"Throughout (e.g., (1.12), Proposition 4, Lemma 5)"},{"comment":"The decay exponents are mutually inconsistent: Theorem 1 defines sigma = 2(1-1/p)-2epsilon, which gives a decay factor (1+t)^{-(1-1/p)+epsilon} in (1.11); the remark after Theorem 1 states the rate as t^{-(1-1/p-epsilon/2)}; and the abstract claims the rate is 'the same as' the two-dimensional heat equation, whose rate is stated as t^{-(1-1/p)}. The theorem's proven rate is the slowest of the three. Please harmonize the statements and qualify the abstract to say 'up to an arbitrarily small epsilon-loss'.","section":"Theorem 1 and the remark following it"},{"comment":"In the definition of the diffuse projection, the integration region 'u3 > 1' should be 'u3 > 0' (the correct version appears in (1.6)); in Definition 2 the velocity set V_n is defined using sign(x1_3) where sign(xn_3) is intended.","section":"Section 1.2 (definition of P_gamma) and Definition 2"},{"comment":"In the statement of Lemma 7 the boundary term '|(1+t)^{sigma/2}(I-P_gamma)f|_{L^2_{T,gamma+}}' appears without the L^1_k norm that the proof produces and that all parallel terms carry; presumably the L^1_k norm is missing from the display.","section":"Lemma 7 statement"}],"recommendation":"major_revision","confidential_remarks":"The paper is long and detailed, and the central estimates are mostly laid out carefully, but the deferred stochastic-cycle lemmas are the real obstruction: I would not recommend acceptance until Lemma 9 and Lemma 17 (or a precise reduction of them to [27] with constants verified in the slab) actually appear in the manuscript, because they are what make the L^infty boundary bootstrap close. I also suggest asking the authors to clarify the relationship to [6], which the introduction describes as a mixed specular/diffuse problem in an infinite-layer-like geometry; the claim to be the 'first' global result for non-compact diffuse boundaries should be worded so that the precise distinction is explicit. Finally, the three different decay exponents (abstract, remark, theorem) should be reconciled before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Top line: this is the first global existence and decay result for the Boltzmann equation with diffuse reflection in a non-compact domain, and the proof is largely real. The Fourier-in-the-horizontal-variable strategy plus the test-function method for the macroscopic part is a genuine adaptation, not a repackaging. The decay rate matching the 2D heat equation is consistent with Kagei's Navier-Stokes results and is compared with, not derived from, that heat equation, so there is no circularity.\n\nWhat the paper does well: it carries the L1_k ∩ Lp_k machinery into a slab with physical boundaries, constructs elliptic test functions with frequency weights to handle the degenerate low-frequency regime, and uses a clever time-derivative bootstrap to recover full dissipation of b3 and c. The 2D problem in R × (-1,1) is handled by a separate L2-L∞ method in physical space, which is the right tool because 1D decay is too slow for the Fourier-side approach. The 60 pages are detailed and the main chain of estimates is internally consistent on inspection.\n\nThe real soft spot is exactly what the stress-test flags: Lemma 9 and Lemma 17, the survival estimates for stochastic cycles after n = C1 T0^{5/4} boundary collisions, are deferred to [27] without proof. These are load-bearing for the L∞ bounds in Proposition 6 and Lemma 16, and without them the multi-reflection boundary terms do not become o(1). In this slab geometry the backward exit time depends only on v3, so adaptation from Guo's bounded-domain paper is plausible, but the quantitative probability bound needs re-deriving; the constants and rates from a 3D bounded domain do not automatically transfer. The referee should be asked to verify this specifically.\n\nA secondary gap is the existence step, dismissed as a standard sequential argument in §4.4 and §6.3. That is common in this literature when the a priori estimates are in hand, but here the nonlinearity and the boundary coupling make it worth a few lines. The notation slips in Lemma 8 are minor and do not affect the argument.\n\nBottom line: this paper deserves serious peer review. The main claims are new and the proof is coherent except for the deferred stochastic-cycle lemmas. I would send it to a referee with the instruction to focus on Lemmas 9 and 17 and the sequential existence construction.","headline":"First global result for diffuse-boundary Boltzmann in non-compact domains; proof is serious but leans on two deferred stochastic-cycle lemmas that should be supplied or referenced precisely.","tokens_in":84163,"tokens_out":1499,"would_cite":true,"duration_ms":18728,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q20","35B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves the first global-in-time solutions for the nonlinear Boltzmann equation confined between two infinite, diffusely reflecting plates, with the decay rate of the two-dimensional heat equation in the three-dimensional slab.","keywords":["Boltzmann equation","infinite layer","diffuse reflection boundary","global in time solutions","large time behavior","polynomial decay","macroscopic dissipation"],"falsifier":"Check the deferred survival estimate directly: compute, analytically or by Monte Carlo simulation for the diffuse reflection kernel in the slab (-1,1), the probability that a backward stochastic cycle survives more than n = C_1 $T_0^{{5/4}}$ collisions for large T_0, and compare with the asserted (1/2)^{C_2 $T_0^{{5/4}}$} decay. A parameter regime where this probability decays only polynomially, or a demonstration that the L^infty bootstrap in Proposition 6 requires a stronger kernel condition than diffuse reflection provides, would void the global-existence claim as proven.","tokens_in":82975,"feed_emoji":"🌀","tokens_out":7887,"duration_ms":69791,"temperature":0.7,"pith_summary":"An isothermal rarefied gas confined between two infinite parallel plates, with walls that scatter particles diffusely, is shown to have unique global-in-time solutions near the Maxwellian state. In the three-dimensional slab the perturbation decays at a polynomial rate equal to that of the two-dimensional heat equation, matching the picture Kagei established for compressible Navier-Stokes in the same geometry. The paper presents this as the first global-existence result for the Boltzmann equation with non-compact, diffuse boundaries. A separate theorem establishes global existence in the two-dimensional slab, where the decay-rate analysis is left open.","feed_headline":"Rarefied gas in a slab decays like 2D heat flow","feed_subtitle":"First global solutions for the Boltzmann equation with diffuse, non-compact walls; the decay rate matches heat conduction.","key_machinery":"The proof Fourier-transforms the equation in the two tangential directions, turning the slab problem into a one-dimensional boundary-value problem on $x_3\\in(-1,1)$ with the frequency $k\\in\\mathbb{R}^2$ as a parameter. The argument combines an $L^1_k\\cap L^p_k$ method in frequency space with the $L^2_{x_3,v}\\cap L^\\infty_{x_3,v}$ interplay technique: the macroscopic (fluid) part is controlled by a dual weak-formulation estimate using elliptic test functions weighted by $|k|^2/(1+|k|^2)$, which produces the frequency-weighted dissipation norm $\\|\\frac{|k|}{\\sqrt{1+|k|^2}}(\\hat a,\\hat b,\\hat c)\\|$, while the kinetic part is controlled by the method of characteristics with stochastic cycles for the diffuse boundary. The critical tool for closing the nonlinearity in $L^\\infty$ is a technical bound (Lemmas 9 and 17) asserting that after $n = C_1 T_0^{5/4}$ boundary collisions, the survival probability of the backward stochastic cycle is at most $(1/2)^{C_2 T_0^{5/4}}$; here the paper defers the proof to an earlier result instead of carrying it out.","core_discovery":"The central claim is that the initial-boundary value problem for the nonlinear Boltzmann equation in the infinite slab with diffuse, isothermal walls admits global solutions close to the Maxwellian. In the three-dimensional slab (Theorem 1), with initial perturbation small in $L^1_k L^\\infty_{x_3,v} \\cap L^p_k L^2_{x_3,v}$ for $2 < p \\le \\infty$, the solution satisfies time-weighted estimates with weight $(1+t)^{\\sigma/2}$, where $\\sigma = 2(1-1/p)-2\\varepsilon > 1$, yielding decay like $t^{-(1-1/p-\\varepsilon/2)}$; the paper emphasizes that this rate is the same as that of solutions to the two-dimensional heat equation. Theorem 2 refines the dissipation estimates for the macroscopic components $\\hat b$ and $\\hat c$, including in the low-frequency regime $|k|\\to 0$, by exploiting the time derivative and Poincar\\'e's inequality. In the two-dimensional slab $\\mathbb{R}\\times(-1,1)$ (Theorem 3), where the decay approach fails because the analogous $\\sigma$ would be below 1, the paper proves global existence of $L^2\\cap L^\\infty$ solutions via a combined estimate on $f$ and $\\partial_t f$. The paper states this is the first result on global solutions of the Boltzmann equation with non-compact and diffuse boundaries.","pith_inferences":["The proofs of the two survival-probability lemmas (Lemma 9 in Section 4.3 and Lemma 17 in Section 6.2) are deferred to an earlier framework, with the text saying 'The proof is similar to [27]' and 'the same as Lemma 9'; a reader building on this theorem should treat those estimates as the part of the argument most in need of independent verification.","The failure of the decay argument in the two-dimensional slab matches an effective-tangential-dimension picture: the same mechanism that gives the $t^{-(1-1/p)}$ rate in $\\mathbb{R}^2$ would give a rate too slow to close the nonlinear estimate in $\\mathbb{R}$, so the open one-dimensional rate, if true, needs a different closing mechanism than the time-weighted norm.","The same machinery, with the time-derivative estimate already developed, is positioned to attack kinetic shear flow (Couette-type) problems where the two plates move tangentially, which the paper notes as a possible next step."],"forward_implications":["In the three-dimensional slab, the solution decays like $t^{-(1-1/p-\\varepsilon/2)}$ in the frequency-integrated norms, so long-time dynamics is governed by tangential diffusion exactly as for the two-dimensional heat equation.","The theorem transfers Kagei's decay picture for compressible Navier-Stokes in an infinite layer to the kinetic (Boltzmann) level in the same geometry.","With a small additional condition on the time derivative of the initial data, the macroscopic components $\\hat b_3$ and $\\hat c$ dissipate without the low-frequency degeneracy, indicating that the fluid part decays like free heat flow even at long wavelength.","In the two-dimensional slab $\\mathbb{R}\\times(-1,1)$, global existence holds under $L^2\\cap L^\\infty$ assumptions together with a time-derivative estimate, but the decay rate is left open."],"supporting_citations":[{"why":"Supplies the L^2-L^infty stochastic-cycle framework for diffuse boundaries; the survival-probability estimates (Lemmas 9 and 17) that the infinite-layer proof borrows are stated to be similar to or the same as results in this work.","marker":"[27]"},{"why":"Supplies the L^1_k cap L^p_k frequency-space method used to obtain time decay and to close the a priori estimate.","marker":"[15]"},{"why":"Supplies the test-function weak formulation for macroscopic dissipation estimates with diffuse reflection, adapted here with the frequency weight |k|^2/(1+|k|^2).","marker":"[18]"},{"why":"Companion to [18]; the same dual weak-formulation argument underlies the macroscopic estimates for hat b and hat c in the slab.","marker":"[19]"},{"why":"Kagei's analysis of the linearized compressible Navier-Stokes semigroup in the infinite layer; provides the two-dimensional heat-equation decay benchmark the paper matches.","marker":"[35]"},{"why":"Kagei's large-time behavior result for Navier-Stokes in the infinite layer; the stated consistency of the decay rate with the present theorem.","marker":"[37]"},{"why":"Guo's global existence for the Boltzmann equation in whole space without time decay; the comparison baseline for why the slab with physical boundary needs new machinery.","marker":"[25]"}],"fun_headline_variants":["Global solutions for Boltzmann in slab with diffuse walls","Rarefied gas slab decays like 2D heat equation","Boltzmann global existence in infinite layer with diffuse boundaries","First proof of global rarefied gas flows in infinite layer","Slab gas dynamics: global existence and heat-like decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the assumption, taken from an earlier paper rather than verified here, that when a particle trajectory is traced backwards and bounces n = C_1 $T_0^{{5/4}}$ times off the diffuse walls, the probability that the path is still alive is at most (1/2) raised to a multiple of $T_0^{{5/4}}$; if that survival decay fails for the diffuse kernel in the slab, the global-existence conclusion does not follow from the argument given.","fun_headline_variants_meta":{"raw":{"variants":["Global solutions for Boltzmann in slab with diffuse walls","Rarefied gas slab decays like 2D heat equation","Boltzmann global existence in infinite layer with diffuse boundaries","First proof of global rarefied gas flows in infinite layer","Slab gas dynamics: global existence and heat-like decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1390,"prompt_tokens":965,"completion_tokens":425,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":344}},"tokens_in":581,"tokens_out":425,"duration_ms":4380,"temperature":1.0,"reasoning_tokens":344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:33:49.874504+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the deferred survival estimate directly: compute, analytically or by Monte Carlo simulation for the diffuse reflection kernel in the slab (-1,1), the probability that a backward stochastic cycle survives more than n = C_1 $T_0^{{5/4}}$ collisions for large T_0, and compare with the asserted (1/2)^{C_2 $T_0^{{5/4}}$} decay. A parameter regime where this probability decays only polynomially, or a demonstration that the L^infty bootstrap in Proposition 6 requires a stronger kernel condition than diffuse reflection provides, would void the global-existence claim as proven.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the L^2-L^infty stochastic-cycle framework for diffuse boundaries; the survival-probability estimates (Lemmas 9 and 17) that the infinite-layer proof borrows are stated to be similar to or the same as results in this work."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the L^1_k cap L^p_k frequency-space method used to obtain time decay and to close the a priori estimate."},{"cited_title":"Esposito, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the test-function weak formulation for macroscopic dissipation estimates with diffuse reflection, adapted here with the frequency weight |k|^2/(1+|k|^2)."},{"cited_title":"Esposito, Y","cited_arxiv_id":null,"evidence_quote":"Companion to [18]; the same dual weak-formulation argument underlies the macroscopic estimates for hat b and hat c in the slab."},{"cited_title":"Kagei, Asymptotic behavior of the semigroup associated with the linearized compressible Navier-Stokes equation in an infinite layer , Publ","cited_arxiv_id":null,"evidence_quote":"Kagei's analysis of the linearized compressible Navier-Stokes semigroup in the infinite layer; provides the two-dimensional heat-equation decay benchmark the paper matches."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Kagei's large-time behavior result for Navier-Stokes in the infinite layer; the stated consistency of the decay rate with the present theorem."},{"cited_title":"Guo, The Boltzmann equation in the whole space , Indiana University Mathematics Journal, 53 (2004), pp","cited_arxiv_id":null,"evidence_quote":"Guo's global existence for the Boltzmann equation in whole space without time decay; the comparison baseline for why the slab with physical boundary needs new machinery."}],"review_version":1}