{"id":"4d21ffe8-acb2-493e-b0f1-508f85705928","arxiv_id":"2504.16560","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For linear Boltzmann transport problems whose data vanish on the inflow and characteristic boundary parts, strong solutions exist with arbitrary spatial Sobolev regularity, including with a continuous slowing-down energy term.","lead":"The paper proves existence and uniqueness of spatially regular strong solutions for a class of linear Boltzmann transport equations in bounded strictly convex domains, provided the data vanish near the inflow and characteristic parts of the boundary. This matters because higher-order Sobolev regularity is normally impossible for these characteristic boundary value problems, and the result clarifies exactly when it can be guaranteed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.19 asserts m-dissipativity of ~Q_C,m(E) without proving the required range condition, and Theorem 3.21 depends on that unproven range surjectivity.","rationale":"I read the theorem statements and proofs in detail, focusing on the chain leading to Theorem 3.21. The paper explicitly assumes strict convexity (Section 2), and Remark 3.5 shows why this is needed for continuity of the extended escape time. This is an honest limitation, not an internal inconsistency, so I do not treat it as a flaw in the central claim as stated. The more serious issue is the unproven range condition in Lemma 3.19. The proof of that lemma demonstrates the dissipativity estimate but does not establish surjectivity of λI - ~Q_C,m(E), which is essential for m-dissipativity and for applying Theorem 3.14. The reader's verdict already flags this as a needed detail and recommends conditional acceptance; my analysis agrees that this is the key soft spot. I therefore do not change the verdict, but I emphasize that the gap is in the proof of the main time-dependent theorem, not merely a cosmetic omission. The concrete test is an analytical re-derivation that would settle whether the range condition holds. If it fails, the paper's central claim would be unproven as written; if it succeeds, the conditional acceptance is justified.","tokens_in":35599,"tokens_out":6572,"duration_ms":63064,"concrete_test":"Write out the full range-condition proof for Lemma 3.19: for f in the dense subspace ∩_{k=1}^∞ C^(k,0)_0(G×S,Γ'_-), define u via the explicit formula u(x,ω) = ∫_0^{t(x,ω)} exp(-∫_0^t â(E)(Σ-K_r+C)(x-sω,ω,E) ds) â(x,E) f(x-tω,ω) dt, and verify (i) u ∈ ∩_{k=1}^∞ C^(k,0)_0(G×S,Γ'_-) using the strict-convexity continuity of the extended escape time (Proposition 3.6) and the support argument of Lemma 3.8; (ii) the H^(m,0)_0-norm of u is controlled by a constant times ||f||_{H^(m,0)_0}; and (iii) (λI - ~Q_C,m(E))u = f in H^(m,0)_0. If all three steps check out, the gap is filled; otherwise Theorem 3.21 requires a different argument or an additional hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central time-dependent result, Theorem 3.21, is obtained by applying the evolution-equation theorem (Theorem 3.14) to the abstract Cauchy problem (95) with generator ~Q_C,m(E). For Theorem 3.14 to apply, each ~Q_C,m(E) must be m-dissipative on H^(m,0)_0(G×S,Γ'_-). Lemma 3.19 is the only result supplying this property. Its proof, however, establishes only the accretivity estimate (83) for S_m(E) (and hence a lower bound for ~Q_C,m(E)+CI), and then states that 'the stated m-dissipativity of ~Q_C,m(E) follows by similar arguments as used in Theorems 3.9 and 3.12 above. We omit further details.' The omitted part is exactly the range condition R(λI - ~Q_C,m(E)) = H^(m,0)_0, which is not a formality: unlike Theorem 3.9, the differential operator here is (1/â(E))ω·∇x, so the explicit solution formula (37) must be re-derived for the equation ω·∇x u + â(E)(Σ - K_r + C)u = â(E)f, and one must verify that the resulting u lies in H^(m,0)_0 and satisfies the support property of Lemma 3.8. The proof does not show these steps. Without the range condition, m-dissipativity is not established, so the existence of φ in Theorem 3.21 is not justified. Lemma 3.15 also omits details of the bound (70), but that is a boundedness estimate and less critical; the range condition is the load-bearing gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves existence of spatially regular strong solutions for a class of linear Boltzmann transport equations on a bounded strictly convex C^∞ domain G ⊂ R^3. The equation is a ∂ψ/∂E + ω·∇_x ψ + Σψ − K_r ψ = f with inflow boundary condition ψ|Γ_- = 0 and, when a ≠ 0, terminal condition ψ(·,·,E_m)=0. The authors work in anisotropic Sobolev spaces H^{(m,0,0)}_0(G×S×I, Γ_-) and H^{(m,0)}_0(G×S, Γ'_-), where functions vanish near the inflow and characteristic parts of the boundary. The main results are Theorem 3.9 (stationary case with a=0, K_r=0), Theorem 3.12 (stationary case with K_r = K_r^2), and Theorem 3.21 (time/energy-dependent case with a<0). The proofs use explicit solution formulas along characteristics for the convection-attenuation equation, a support-propagation lemma for the escape-time map, m-accretivity/closed-extension arguments for the spatial operator, and the abstract evolution-equation theorem of Tanabe/Pazy for the energy-dependent case.","tokens_in":35947,"tokens_out":4323,"duration_ms":42882,"significance":"If the results are fully established, they provide a clean positive answer to a subtle question: although full isotropic Sobolev regularity fails for characteristic transport problems with variable boundary multiplicity, regularity in anisotropic spaces with data vanishing on the inflow and characteristic boundary can be recovered. The explicit formula (37) and the support propagation Lemma 3.8 are convincing and are applied in a logically clear way for Theorems 3.9 and 3.12. The paper also carefully identifies the geometric role of strict convexity (Proposition 3.6 and Remark 3.5), which is an instructive and honest limitation. The dependence on the authors' earlier work for trace theorems, Green's formula, and L^2 well-posedness is explicit and appears non-circular, since those results are already published and do not presuppose the current theorems. The main weakness is that the proof of the central time-dependent result (Theorem 3.21) relies on an unproven range-condition claim in Lemma 3.19.","major_comments":[{"comment":"Lemma 3.15 states the bound (70) for ‖(1/â(E)) K̂_r(E)‖_m and says 'We omit further details.' This bound is used in the definition of the operator family and in the proof of Lemma 3.19, and it is also needed to justify the C^1-dependence of the family in Theorem 3.21. The estimate is plausible and likely derivable by the methods of Theorem 3.11, but an explicit proof or a precise reference should be included, especially because the proof of Lemma 3.19 already leaves a critical gap.","section":"§3.3, Lemma 3.19 and Theorem 3.21"}],"minor_comments":[{"comment":"In the sentence 'the induction hypothesis implies that ψ ∈ C^{(m,0,0)}_0(G×S×, Γ_-)', the domain 'G×S×' is missing the factor I; it should read 'G×S×I'.","section":"§3.1, Theorem 3.9 proof, Part A.2"},{"comment":"The notation 'Γ_- ∪ Γ_0 = Γ_-' is an abuse, since Γ_0 is not a subset of Γ_-; the intended meaning is that Γ_- ∪ Γ_0 is the closure of Γ_- in Γ. This shorthand recurs in several places and could confuse readers unfamiliar with the boundary decomposition.","section":"§2 and §3.1, Lemma 3.8"},{"comment":"The space denoted W^{∞,(m,0,0)}(G×S×I) is redefined as a completion of C^{(∞,0,0)}(G×S×I) after the notation was already used for the L^∞-based Sobolev space in (19). Using the same symbol for two different spaces is confusing; a distinct notation would be clearer.","section":"§3.1.1"},{"comment":"The proof of Theorem 3.21 invokes several prior results from the authors' earlier papers ([34], [35], [36], [40]). A short table or index identifying exactly which prior theorem is used at each step would make the dependence more transparent and ease verification.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The omitted range condition in Lemma 3.19 is a genuine load-bearing gap for the paper's headline result, Theorem 3.21. I do not see an obvious contradiction—the result may well be true and provable by adapting the explicit-formula arguments—but the proof as written is incomplete. The authors should also provide the missing proof of Lemma 3.15. Given that the central claim is defensible but needs substantial additional argumentation, major revision is appropriate rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before reading this paper. First, it delivers what it claims: existence of strong solutions with arbitrary order m in the spatial variable for linear transport problems (attenuation, scattering, and continuous slowing-down) on bounded strictly convex domains, assuming the data vanish on the inflow and characteristic boundary. That is a real step beyond existing L2 and low-regularity results, and the connection to discontinuous Galerkin convergence rates is credible. Second, the paper has a genuine gap: Lemma 3.19 asserts m-dissipativity for the generator in the CSDA case without proving the range condition, and Theorem 3.21 leans on that. The reader's report and the stress-test note are right on this point.\n\nWhat is actually new: the combination of zero-trace anisotropic Sobolev spaces with the support condition on σ2 to make the scattering operator bounded into the zero-trace space, and the use of explicit solution formulas plus closures to get range-surjectivity for the stationary operators. The proof of Theorem 3.9 (convection-attenuation) is carefully written and the induction on m is natural. Lemma 3.8's support propagation argument is clean and uses the continuity of the extended escape time in an essential way. Theorem 3.12's reduction to m-accretivity via a bounded perturbation is standard but correctly handled.\n\nWhere it gets soft: Lemma 3.19 says \"the stated m-dissipativity follows by similar arguments as used in Theorems 3.9 and 3.12 above. We omit further details.\" That is not a formality. For m-dissipativity you need range R(λI − ~Q_C,m(E)) to be the whole space, and here the transport operator has the factor 1/â(E) multiplying ω·∇x, so the explicit solution formula and the support property have to be re-verified for that scaled operator. The paper does not show it. The stress-test note is accurate: without it, Theorem 3.21 does not follow. Lemma 3.15 also omits details, but that is a boundedness estimate and less worrying. The strict convexity assumption is honestly flagged as load-bearing in Remark 3.5 and Proposition 3.6, so that is not a hidden flaw.\n\nThe work relies on the authors' own prior results for trace theorems, Green's formula, and the C1 escape time. Those are published and independent, so the citation pattern is fine. No circularity problem.\n\nBottom line: the main existence theorems are plausible, and the stationary parts are essentially proven. The CSDA theorem needs the missing range-condition proof. I would send this to a competent referee, not desk reject, but the referee should be asked to check Lemma 3.19 carefully. If the range condition fills in, this is a solid paper for transport regularity and numerical analysis. If it doesn't, Theorem 3.21 is unproven as stated. Worth engaging with.","headline":"Plausible, genuinely new high-order spatial regularity for transport equations, but the time-dependent theorem currently rests on an unwritten range-condition proof.","tokens_in":36470,"tokens_out":2936,"would_cite":true,"duration_ms":26041,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q20","35L04","35B65","47D06"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a bounded strictly convex domain, the paper proves that linear Boltzmann transport problems with data supported away from the inflow and grazing boundary have unique strong solutions with arbitrary finite spatial Sobolev regularity…","keywords":["linear Boltzmann transport equation","initial inflow boundary value problem","characteristic boundary","variable multiplicity","anisotropic Sobolev spaces","escape-time map","spatial regularity","continuous slowing down approximation"],"falsifier":"Take the unit ball $G=B(0,1)$ with $\\Sigma=0$, $C=0$, and a smooth $f$ supported away from $\\Gamma_-\\cup\\Gamma_0$; the explicit formula and the continuity of $\\tilde t(x,\\omega)=x\\cdot\\omega+\\sqrt{(x\\cdot\\omega)^2+1-\\|x\\|^2}$ predict $\\psi\\in H^{(m,0,0)}_0$ for every $m$. Repeat the same construction on a domain with a flat boundary piece, such as a cylinder or cube: by Remark 3.5 the extended escape-time map is discontinuous at grazing directions over the flat face, and a direct calculation of the first-order derivatives from formula (39) should exhibit a jump there, giving $\\psi\\notin H^{(2,0,0)}$ even for smooth vanishing data. If instead the solution remains $H^{(2,0,0)}$ on the flat-faced domain, then strict convexity is not essential; if it fails, the theorem's domain hypothesis is necessary for the proof's mechanism.","tokens_in":35385,"feed_emoji":"⚛️","tokens_out":14570,"duration_ms":127739,"temperature":0.7,"pith_summary":"The paper establishes that the obstruction to higher-order spatial regularity in characteristic transport boundary value problems disappears when the data are required to vanish on the inflow and grazing parts of the boundary. These problems are characteristic with variable multiplicity, meaning the rank of the boundary matrix changes at the grazing set, and unrestricted data are known to give at most $s<3/2$ spatial Sobolev regularity. For the stationary convection-scattering equation, every datum $f$ in the anisotropic space $H^{(m,0,0)}_0(G\\times S\\times I,\\Gamma_-)$ yields a unique strong solution in the same space, for every finite order $m$ and for a strictly convex $C^\\infty$ domain $G\\subset\\mathbb R^3$. For the continuous-slowing-down equation with coefficient $a$ satisfying $-a\\ge\\kappa>0$, any $f\\in C^1(I,H^{(m,0)}_0(G\\times S,\\Gamma'_-))$ produces a unique solution $\\psi\\in C^1(I,H^{(m,0)}_0(G\\times S,\\Gamma'_-))$ with zero inflow data and zero value at the cutoff energy. The paper's message is that this limited regularity is a phenomenon of unrestricted data, not an intrinsic obstruction: vanishing near the inflow and grazing boundaries restores full finite-order spatial smoothness.","feed_headline":"Inflow-vanishing data restore full Sobolev regularity in transport","feed_subtitle":"Zero data near inflow and grazing boundaries yield unique C^1 strong solutions at any spatial order.","key_machinery":"The argument rests on three pieces. First, the anisotropic Sobolev spaces $H^{(m,0,0)}_0(G\\times S\\times I,\\Gamma_-)$ — completions of smooth functions whose support avoids $\\Gamma_-\\cup\\Gamma_0$ — encode exactly the vanishing data condition that makes higher $x$-regularity possible. Second, the extended escape-time map $\\tilde t:\\overline G\\times S\\to\\mathbb R_+$, defined as the time to reach $\\Gamma_-$ along the backward characteristic, is continuous on $\\overline G\\times S$ because $G$ is strictly convex; this continuity is what allows the explicit solution formula $\\psi(x,\\omega,E)=\\int_0^{t(x,\\omega)}e^{-\\int_0^t\\Sigma(x-s\\omega,\\omega,E)\\,ds}f(x-t\\omega,\\omega,E)\\,dt$ to be differentiated in $x$ and to propagate the support condition. Third, the smallest closed extension of the transport operator is shown to be $m$-accretive, a monotonicity-plus-range property, and surjective, so the stationary problem is well posed in every finite-order space; the scattering operator $K_r$ enters as a bounded perturbation, and in the energy-dependent case the family of operators has $E$-independent domain, allowing an abstract evolution-equation theorem to convert $m$-dissipativity into a $C^1$-in-$E$ solution.","core_discovery":"The central claim is Theorem 3.21: under the hypotheses $\\Sigma\\in C^1(I,W^{\\infty,(m,0)}(G\\times S))$, $a\\in C^1(I,W^{\\infty,m}(G))$, $-a\\ge\\kappa>0$, $\\sigma_2$ vanishing near $\\Gamma_-\\cup\\Gamma_0$ in the appropriate Sobolev sense, and $f\\in C^1(I,H^{(m,0)}_0(G\\times S,\\Gamma'_-))$, the initial inflow problem $a\\,\\partial\\psi/\\partial E+\\omega\\cdot\\nabla_x\\psi+\\Sigma\\psi-K_r\\psi=f$, $\\psi|_{\\Gamma_-}=0$, $\\psi(\\cdot,\\cdot,E_m)=0$ has a unique strong solution $\\psi\\in C^1(I,H^{(m,0)}_0(G\\times S,\\Gamma'_-))$. The companion stationary result, Theorem 3.12, asserts that the time-independent problem $\\omega\\cdot\\nabla_x\\psi+\\Sigma\\psi-K_r\\psi+C\\psi=f$, $\\psi|_{\\Gamma_-}=0$, has a unique solution $\\psi\\in H^{(m,0,0)}_0(G\\times S\\times I,\\Gamma_-)$ for every $f$ in that space. Here $K_r$ is the restricted collision operator $K_r\\psi=\\int_{S'}\\sigma_2(x,\\omega',\\omega,E)\\psi(x,\\omega',E)\\,d\\omega'$, and the subscript $0$ in the function space means the function vanishes on the inflow boundary and on the characteristic (grazing) set in the Sobolev sense. The statement is the author's intended contribution: full spatial regularity is available for these characteristic, variable-multiplicity problems once the data live in the anisotropic spaces that vanish on $\\Gamma_-\\cup\\Gamma_0$.","pith_inferences":["The strict-convexity hypothesis is likely not sharp: what the proof needs is that the extended escape-time map $\\tilde t$ be continuous up to the boundary, so domains whose boundary has no flat pieces should suffice; a direct check of the unit ball formula $\\tilde t(x,\\omega)=x\\cdot\\omega+\\sqrt{(x\\cdot\\omega)^2+1-\\|x\\|^2}$ shows continuity there, while the paper's own Remark 3.5 shows flat faces b","The same accretivity-plus-bounded-perturbation route should extend to the full restricted collision operator $K_r=K^1_r+K^2_r+K^3_r$ (including energy-transfer and straggling terms) whenever each cross-section vanishes near $\\Gamma_-\\cup\\Gamma_0$ in the analogous Sobolev sense; the paper carries out only the $K^2_r$ case.","A numerical experiment could separate the regularity obstruction from discretization error: approximate a stationary transport problem on a ball with boundary-adapted finite elements enforcing zero values near $\\Gamma_-\\cup\\Gamma_0$, and compare the observed convergence order with $h^{k+1/2}$; the vanishing-data theory predicts the higher rate, whereas unrestricted data would stall near $h^{1/2}$.","If the strict-convexity premise is essential rather than technical, a cube or cylinder should exhibit loss of the claimed $H^m$ regularity even for smooth data vanishing away from the boundary; computing the explicit solution on such a domain would locate exactly which geometric feature the result needs."],"forward_implications":["For stationary problems whose data vanish on $\\Gamma_-\\cup\\Gamma_0$, solutions inherit the full finite spatial Sobolev order $m$ of the data; the known restriction $s<3/2$ for unrestricted data does not apply in this vanishing-data setting.","For the continuous-slowing-down equation, the solution is $C^1$ in the energy variable as a map into $H^{(m,0)}_0(G\\times S,\\Gamma'_-)$, giving a Hilbert-space-valued strong solution with no loss of $x$-regularity.","The range equality $R(\\tilde Q_m+CI)=H^{(m,0,0)}_0(G\\times S\\times I,\\Gamma_-)$ holds for every finite $m$, so the regularity bottleneck in discontinuous Galerkin convergence estimates can be removed: error rates such as $h^{\\min\\{r,k+1\\}-1/2}$ are limited by the polynomial degree $k$ rather than by the solution's Sobolev index $r$.","Nonzero inflow data $g$ are handled by an explicit lift: when the transformed source in (52) stays in the vanishing space, the solution splits as an $H^{(m,0,0)}_0$-part plus a lift $L_-g$, with the compatibility condition $g(\\cdot,\\cdot,E_m)=0$ controlling the energy regularity.","The abstract accretivity framework also yields compatibility conditions (Remark 3.24) that must hold for higher-order $E$-regularity, with zeroth order being sufficient for the $m_3=1$ case proved here."],"supporting_citations":[{"why":"Supplies the characteristic initial-boundary-value framework with restricted data whose vanishing-space formulation this paper adapts to transport equations.","marker":"[26]"},{"why":"Provides the inflow trace theorem, Green formula, and explicit solution formula for transport problems that the paper differentiates.","marker":"[35]"},{"why":"Establishes the escape-time regularity result for convex domains and the trace-space background, plus the counterexample limiting unrestricted spatial regularity.","marker":"[34]"},{"why":"Supplies the abstract evolution-equation theorem used to convert m-dissipativity into the C^1-in-energy solution of Theorem 3.21.","marker":"[33]"},{"why":"Gives the criterion that an accretive operator is m-accretive iff its range is the whole space, used to prove the surjectivity of the closed transport operators.","marker":"[28]"},{"why":"Provides the accretivity estimate for minus inverse-a times the streaming operator (Lemma 3.17) and the CSDA problem framework used in Section 3.3.","marker":"[36]"},{"why":"Supplies fundamental existence, uniqueness, and lift results for inflow transport problems on which the strong-solution argument builds.","marker":"[10]"},{"why":"Supplies the continuity lemma for the escape-time map invoked in Proposition 3.6 under strict convexity.","marker":"[5]"}],"fun_headline_variants":["Full spatial regularity for transport with inflow-vanishing data","Vanishing data on inflow and grazing boundaries give full regularity","Anisotropic Sobolev spaces rescue regularity in transport problems","Zero boundary data unlock full Sobolev regularity for transport"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is strict convexity of the bounded $C^\\infty$ domain $G$: it makes the extended escape-time map $\\tilde t$ continuous on $\\overline G\\times S$, and that continuity is what lets the proof differentiate the explicit solution formula and propagate the data's vanishing near the inflow and grazing boundary.","fun_headline_variants_meta":{"raw":{"variants":["Full spatial regularity for transport with inflow-vanishing data","Vanishing data on inflow and grazing boundaries give full regularity","Anisotropic Sobolev spaces rescue regularity in transport problems","Zero boundary data unlock full Sobolev regularity for transport"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000358,"raw_usage":{"total_tokens":1980,"prompt_tokens":1023,"completion_tokens":957,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":901}},"tokens_in":639,"tokens_out":957,"duration_ms":7407,"temperature":1.0,"reasoning_tokens":901,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:00:58.716116+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the unit ball $G=B(0,1)$ with $\\Sigma=0$, $C=0$, and a smooth $f$ supported away from $\\Gamma_-\\cup\\Gamma_0$; the explicit formula and the continuity of $\\tilde t(x,\\omega)=x\\cdot\\omega+\\sqrt{(x\\cdot\\omega)^2+1-\\|x\\|^2}$ predict $\\psi\\in H^{(m,0,0)}_0$ for every $m$. Repeat the same construction on a domain with a flat boundary piece, such as a cylinder or cube: by Remark 3.5 the extended escape-time map is discontinuous at grazing directions over the flat face, and a direct calculation of the first-order derivatives from formula (39) should exhibit a jump there, giving $\\psi\\notin H^{(2,0,0)}$ even for smooth vanishing data. If instead the solution remains $H^{(2,0,0)}$ on the flat-faced domain, then strict convexity is not essential; if it fails, the theorem's domain hypothesis is necessary for the proof's mechanism.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the characteristic initial-boundary-value framework with restricted data whose vanishing-space formulation this paper adapts to transport equations."},{"cited_title":"On Existence of $L^2$-solutions of Coupled Boltzmann Continuous Slowing Down Transport Equation System","cited_arxiv_id":"1603.05534","evidence_quote":"Provides the inflow trace theorem, Green formula, and explicit solution formula for transport problems that the paper differentiates."},{"cited_title":"On Existence of $L^1$-solutions for Coupled Boltzmann Transport Equation and Radiation Therapy Treatment Optimization","cited_arxiv_id":"1406.3228","evidence_quote":"Establishes the escape-time regularity result for convex domains and the trace-space background, plus the counterexample limiting unrestricted spatial regularity."},{"cited_title":"Equations of Evolution , Pitman, 1979","cited_arxiv_id":null,"evidence_quote":"Supplies the abstract evolution-equation theorem used to convert m-dissipativity into the C^1-in-energy solution of Theorem 3.21."},{"cited_title":"Semigroups of Linear Operators and Applications to Partial Differential Equations , Springer, 1983","cited_arxiv_id":null,"evidence_quote":"Gives the criterion that an accretive operator is m-accretive iff its range is the whole space, used to prove the surjectivity of the closed transport operators."},{"cited_title":", Frank, M","cited_arxiv_id":null,"evidence_quote":"Provides the accretivity estimate for minus inverse-a times the streaming operator (Lemma 3.17) and the CSDA problem framework used in Section 3.3."},{"cited_title":"Mathematical Analysis and Numerical Methods for Science an d Technology, Vol","cited_arxiv_id":null,"evidence_quote":"Supplies fundamental existence, uniqueness, and lift results for inflow transport problems on which the strong-solution argument builds."},{"cited_title":"S., Kovtanyuk, A","cited_arxiv_id":null,"evidence_quote":"Supplies the continuity lemma for the escape-time map invoked in Proposition 3.6 under strict convexity."}],"review_version":1}