{"id":"5d7bf45c-af4d-4ca8-847e-d52ce66921cc","arxiv_id":"2608.04402","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Small-data solutions of the three-dimensional Coulomb Vlasov-Hartree system scatter with logarithmic corrections determined by the opposite species, with explicit asymptotic profiles.","lead":"The paper proves global well-posedness and exact long-time scattering for a coupled Vlasov-Hartree model with Coulomb interaction. For small, well-localized data, the fermionic distribution and the bosonic wave both acquire logarithmic corrections generated by the other species, and the paper identifies the asymptotic profiles.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 assumes data satisfying (1.7), which do not imply the C^1 regularity required by Proposition 3.1, and no approximation argument is supplied.","rationale":"The reader's weakest assumption correctly identifies the mismatch between the C^1 requirement in Proposition 3.1 and the hypotheses of Theorem 1.1. I agree this is the most load-bearing concern because local well-posedness is the entry point for the entire bootstrap and continuation argument; without a local solution in the asserted class, the global theorem and its modified-scattering conclusions are not justified for the full data set stated in (1.7). The issue is concrete and easily demonstrated by a discontinuous-gradient example that satisfies (1.7), and the Vlasov transport structure guarantees the singularity survives, so the failure is not merely a technicality about a clever choice of representative. I do not elevate this to a rejection because the fix is straightforward: either strengthen (1.7) by adding f0 ∈ C^1, which is compatible with the local theory, or replace the C^1 conclusion by the natural W^{1,∞} ∩ W^{1,2} regularity and provide a density argument. The asymptotic laws themselves are supported by the paper's own estimates for the Galilean vector fields, the effective-potential comparisons, and the profile convergence, so the main mathematical mechanism does not appear threatened. I only partially agree with the reader's placement of the unpublished companion [25] as a second core weakness: the current manuscript states and proves the needed Schrödinger-profile estimates directly, so [25] seems to be a methodological antecedent rather than a repository of unstated theorems. Nonetheless, clarifying the role of [25] would improve auditability. On balance, the reader's conditional verdict is appropriate: acceptance should require the regularity hypothesis to be aligned with the local theory.","tokens_in":26490,"tokens_out":48654,"duration_ms":475139,"concrete_test":"Construct an admissible f0 satisfying (1.7) but not C^1, for example f0(x,v)=φ(x)⟨v⟩^{-4} min(|v_1|,1) with φ compactly supported. Then ∥⟨x⟩^4⟨v⟩^4 f0∥_{L∞}, ∥∇f0∥_{L^2∩L^∞}, and ∥|x|∇f0∥_{L∞} are finite, but ∇f0 is discontinuous. Since the Vlasov solution evolves by composition with the characteristic flow, the same discontinuity persists for every time, so f(t) is never C^1. This disproves the regularity conclusion of Theorem 1.1 as written. The fix is verifiable by adding f0 ∈ C^1 to (1.7) and confirming that every subsequent estimate in Sections 3–4 closes verbatim with this additional hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem claims a unique global strong solution with f ∈ C^1([0,∞)×R^6) under assumption (1.7). But Proposition 3.1, the local well-posedness input, requires f0 ∈ C^1(R^6) and proves f ∈ C^1 on the local interval. Assumption (1.7) only controls ∥∇_{x,v}f0∥_{L^2∩L^∞} and weighted bounds; such weak derivative bounds do not imply continuity of the derivative. Thus the hypotheses of Theorem 1.1 do not put the initial data in the class required by the local theory, and the proof provides no approximation of f0 by C^1 data. Moreover, the Vlasov equation is transport along characteristics, so a derivative singularity in f0 persists for all times; the C^1 regularity of f(t) cannot be manufactured by the evolution. This is a genuine gap in the theorem as stated. It is not fatal to the asymptotic mechanism: the bootstrap and modified-scattering estimates use only L^2∩L^∞ derivative bounds and weighted ∞-norms, so adding f0 ∈ C^1 to (1.7) would close the argument immediately, or one could weaken the conclusion to W^{1,∞} ∩ W^{1,2} for the Vlasov component. The separate concern about the unpublished companion [25] is less decisive: the paper appears to reproduce the needed physical-space Hartree estimates in its own propositions rather than importing an unstated theorem as a black box, though the dependence of the method on [25] still warrants clarification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the three-dimensional Vlasov--Hartree system with Coulomb interaction, in which the Vlasov force is generated by the bosonic density |u|^2 and the Hartree potential by the fermionic density \\int f^2 dv. The main theorem (Theorem 1.1) asserts that sufficiently small, localized, and regular data produce a unique global strong solution and that both components exhibit modified scattering: f(t, x+tv-\\log t F_\\infty(v), v) -> f_\\infty in L^\\infty, while u(t,x) is asymptotic to t^{-3/2} e^{-i|x|^2/4t} e^{-i\\log t \\phi_\\infty(x/t)} h_\\infty(x/t). Theorem 1.2 states a short-range analogue with ordinary scattering for interactions |x|^{-\\alpha}, 1<\\alpha<3/2. The proof couples a bootstrap for the Vlasov free-transport profile \\gamma with Galilean vector-field estimates for the Hartree component and derives the asymptotic profiles by constructing limits rather than assuming them.","tokens_in":26767,"tokens_out":16479,"duration_ms":184330,"significance":"If the main theorem is correct, this is the first sharp long-time asymptotic description for this coupled kinetic-dispersive model, and it shows that the two components interact at leading order through logarithmic phase and characteristic corrections. The proof is largely self-contained and contains parameter-free constructions: the asymptotic profiles are limits of the evolution, not fitted objects, and the Lorentz-space estimates used are mostly standard. The extension to short-range interactions is a useful additional result. However, the regularity gap in Theorem 1.1, the sign/phase inconsistency in Section 4.2, and the incomplete energy estimates in Section 3.2 must be fixed before the claims are fully supported.","major_comments":[{"comment":"The hypotheses of Theorem 1.1 do not imply the regularity required by the local well-posedness theory. Assumption (1.7) controls \\nabla_{x,v} f0 only in L^2 \\cap L^\\infty and with a weight in L^\\infty; this does not imply that f0 is C^1, whereas Proposition 3.1 requires f0 \\in C^1 and yields f \\in C^1 on a short interval. Since the proof neither approximates f0 by C^1 data nor supplies a separate local existence argument under (1.7), the global C^1 statement in Theorem 1.1 is not justified. This is a genuine gap in the theorem as stated, although it is localized: adding f0 \\in C^1 to (1.7), or weakening the conclusion for f to W^{1,\\infty} \\cap W^{1,2} in (x,v), would close it.","section":"Section 1, Theorem 1.1; Section 3.1, Proposition 3.1"},{"comment":"The proof of the Hartree asymptotic is inconsistent with the definition of h_\\infty and with the sign of the phase in (1.10). Since h(t,y)=g(t,y)e^{i\\theta(t,y)} and \\theta(t,y)=l(y)+\\log(t)\\phi_\\infty(y)+O(t^{-1/40}), the correct relation is g(t,y)=h(t,y)e^{-i\\theta(t,y)}, so g converges to h_\\infty(y)e^{-i(l(y)+\\log(t)\\phi_\\infty(y))}. The displayed identity in the text instead writes g(t,y)-h_\\infty(y)e^{+i\\lambda(\\log(t)\\phi_\\infty(y)+l(y))} and concludes convergence to the plus-phase profile. Consequently, as written, the proof yields a different phase than the e^{-i\\log(t)\\phi_\\infty} in (1.10), and the l(y) phase is also missing from (1.10). This is fixable by redefining the final h_\\infty to absorb e^{-il(y)} and by correcting the signs throughout Section 4.2, but the current text does not support the stated law.","section":"Section 4.2, Eqs. (4.3)-(4.7) and the paragraph preceding (1.10)"},{"comment":"The energy estimates for Gu and G^2u omit the potential term proportional to \\phi. From (i\\partial_t-\\Delta)Gu=\\phi Gu+2t(\\nabla\\phi)u, the standard identity gives \\partial_t\\|Gu\\|_{L^2} \\le \\|\\phi\\|_{L^\\infty}\\|Gu\\|_{L^2}+2t\\|\\nabla\\phi\\|_{L^\\infty}\\|u\\|_{L^2}. The lemma states only the second term. The missing term is handled by Gr\\\"onwall using Corollary 3.4 (\\|\\phi\\|_{L^\\infty} \\lesssim \\varepsilon^2\\langle t\\rangle^{-1}), which inserts an additional factor of t^{O(\\varepsilon^2)} or \\log^{O(\\varepsilon^2)}; for \\varepsilon small this is compatible with the bootstrap, but the estimate as printed is not justified. The same omission occurs in the G^2 equation in Lemma 3.7. Please provide the corrected inequality and verify that the log powers absorb the extra factor.","section":"Section 3.2, Lemmas 3.5 and 3.7"}],"minor_comments":[{"comment":"The definition of F_\\infty(y) has a typo: it integrates with dy and uses h_\\infty(y) inside a convolution in x. The intended formula is F_\\infty(y)=\\lambda\\int (x-y)/|x-y|^3 |h_\\infty(x)|^2 dx, as used in Corollary 4.4.","section":"Eq. (4.16)"},{"comment":"The paper should state explicitly which estimates from the unpublished companion [25] are actually used as inputs. The proof appears self-contained, but Section 1.2 describes [25] as the driving force; the reader needs a precise dependency list to audit the argument.","section":"Section 1.2 and references"},{"comment":"There are several typographical errors: 'SCA TTERING' in the title, 'gives' in Lemma 2.1(2), and 'femionic' in Lemma 3.7.","section":"Title, Lemma 2.1, and Lemma 3.7"},{"comment":"The chain-rule identity '\\nabla_x\\gamma(t,x-tv,v)=t^{-1}\\nabla_v\\gamma(t,x-tv,v)-t^{-1}\\nabla_v[\\gamma(t,x-tv,v)]' is confusing; please rewrite it with the intended variable change and clarify how integration by parts in v is used.","section":"Section 3.3, Lemma 3.9"},{"comment":"In the final Hartree step, the rate t^{-1/80} is stated without specifying the choice of m \\in (3/2,2) and \\kappa in Proposition 4.3; please spell out the choice so the reader can verify the exponent.","section":"Section 4.2 and Proposition 4.3"}],"recommendation":"major_revision","confidential_remarks":"Not for the authors: the manuscript leans explicitly on the unpublished companion [25]; if the journal is concerned about reproducibility, it may be worth requesting a written confirmation from the authors about which results in [25] are used as inputs. I do not see a novelty-disclosure problem: the asymptotic mechanism is distinctly coupled, and the short-range extension is more than a verbatim transplant. The two main technical issues are correctable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper proves the first modified scattering theorem for the 3D Coulomb Vlasov–Hartree system, and the crossed log corrections (fermionic profile determines the bosonic phase, bosonic profile determines the fermionic drift) are genuinely new. The main bootstrap is detailed and internally coherent. But Theorem 1.1 as stated has a real regularity hole: assumption (1.7) does not imply f0 in C^1, while the local theory in Proposition 3.1 requires C^1 data, and the proof supplies no approximation argument. This is patchable in a paragraph (add f0 in C^1 to (1.7), or weaken the conclusion), and the asymptotic machinery itself only uses L^2∩L^∞ derivative bounds, so the main result is not in danger.\n\nWhat is new: the coupled asymptotic pair (f∞, h∞) with mutually determined log corrections. This is not present in the Vlasov–Poisson modified scattering literature (Ionescu–Pausader–Wang–Widmayer et al.) or in Hartree long-range theory. The physical-space method from the companion [25] is adapted and mostly reproduced; the paper's own propositions (3.2–3.10, 4.1–4.7) carry the estimates, so [25] functions more as a methodological source than an unstated black box. Still, the authors should confirm which estimates are imported verbatim, since an unpublished dependency should be flagged.\n\nSoft spots, in proportion:\n\n1. The regularity gap (the stress-test has it right). A non-C^1 f0 with only weak L^2∩L^∞ derivative bounds will not become C^1; Vlasov transport preserves the singularity. The theorem's proof as written does not produce f in C^1. Fix: add f0 in C^1 to (1.7), or state the conclusion in W^{1,∞}∩W^{1,2} for the Vlasov component. Minor fix, but it must be made.\n\n2. The dependence on [25]. The paper calls it \"the driving force.\" A referee cannot fully audit that input if key estimates are only proved there. The present text seems to prove the needed Hartree asymptotics in Section 4, but the authors should state explicitly which lemmas are original here and which are quoted from [25]. A clarity issue, not a correctness crisis.\n\n3. Small technical points: the abstract has a typo (\"SCA TTERING\"), and the transition from the local continuation criterion to the global logarithmic growth is terse but fine. I did not check every Lorentz-space constant; the estimates look consistent, and I found no contradiction in the main bootstrap.\n\nWho it is for: anyone working on kinetic–dispersive coupling, modified scattering, or Vlasov–Poisson/Hartree asymptotics. It deserves a serious referee. I would send it out, with a request to patch the regularity statement and clarify the [25] dependence.\n\nRecommendation: accept with revision, after the authors fix the f0 regularity hypothesis and state the provenance of the [25] estimates.","headline":"First modified scattering for the coupled Coulomb Vlasov–Hartree system, with a genuine but easily patched regularity gap in the main theorem.","tokens_in":27333,"tokens_out":2664,"would_cite":true,"duration_ms":28972,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q83","35Q55","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the Coulomb Vlasov–Hartree system, a classical–quantum model of a Bose–Fermi mixture, has global smooth solutions for small localized data and that its long-time behavior is modified scattering in which the…","keywords":["Vlasov–Hartree system","Coulomb interaction","modified scattering","long-time asymptotics","Bose–Fermi mixture","global well-posedness","Galilean vector fields","logarithmic phase correction"],"falsifier":"Solve (1.1) numerically in three dimensions for small Gaussian data with $V(x)=\\lambda|x|^{-1}$ and check whether $\\tilde\\gamma(t,x,v)=\\gamma(t,x-\\log(t)F_\\infty(v),v)$ converges in $L^\\infty_{x,v}$ with the stated $t^{-\\delta_0}$ rate and whether the rescaled Schrödinger profile, after removing the phase $e^{-i\\log(t)\\phi_\\infty(x/t)}$, converges in $L^\\infty_x$; failure of either for arbitrarily small data would refute Theorem 1.1. Alternatively, finding data satisfying (1.7) but not $C^1$ for which the system has no strong solution would refute the theorem's regularity statement.","tokens_in":26251,"feed_emoji":"⚛️","tokens_out":9750,"duration_ms":102662,"temperature":0.7,"pith_summary":"This paper studies the Vlasov–Hartree system in three dimensions with Coulomb interaction, a mean-field model of a Bose–Fermi mixture in which the bosonic density generates a force on the fermions and the fermionic density generates a potential for the bosons. For sufficiently small, regular, and localized initial data it proves global well-posedness and, more importantly, identifies the exact long-time asymptotics. The central claim is modified scattering with the coupling visible at leading order: the fermionic distribution converges along characteristics shifted by $\\log(t) F_\\infty(v)$, where $F_\\infty$ is built from the limiting bosonic profile, while the bosonic wave acquires the phase $e^{-i \\log(t)\\phi_\\infty(x/t)}$, with $\\phi_\\infty$ built from the limiting fermionic density. A sympathetic reading is that the two components never decouple: each species' memory of the other survives in the asymptotic data at order one.","feed_headline":"Coulomb Bose–Fermi mixture scatters with log corrections","feed_subtitle":"Small Coulomb data give global solutions, and each species' long-time drift or phase is set by the other.","key_machinery":"The proof is carried out in the common self-similar variable $y=x/t$. For the Coulomb potential it uses an exact shell decomposition $1/|z|=C_0\\int_0^\\infty \\chi(z/R)\\,dR/R^2$, which lets the physical field $E(t,x)$ be compared with an effective field $E^0(t,x/t)$ whose evolution can be controlled through the transport equation for the free-transport profile $\\gamma(t,x,v)=f(t,x+tv,v)$. On the Schrödinger side the central objects are the Galilean vector field $G=ix+2t\\nabla$ and the renormalized profile $h(t,y)=t^{3/2}e^{i|y|^2t/4}u(t,ty)e^{i\\theta(t,y)}$, where $\\theta$ removes the growing Coulomb phase. The key structural fact is that both components propagate in the same variable: convergence of $h$ to $h_\\infty$ gives the limiting force $F_\\infty=-\\nabla(V*|h_\\infty|^2)$ that enters the Vlasov characteristics, and convergence of $\\gamma$ gives the density $\\rho_\\infty$ that defines the phase $\\phi_\\infty=V*\\rho_\\infty$ in the Hartree law.","core_discovery":"With $V(x)=\\lambda|x|^{-1}$, the paper proves (Theorem 1.1) that for initial data satisfying the smallness condition (1.7) there is a unique global strong solution, and that as $t\\to\\infty$ the free-transport profile obeys $f(t, x+tv-\\log(t)F_\\infty(v), v) = f_\\infty(x,v) + O_{L^\\infty_{x,v}}(t^{-\\delta_0})$, while the Hartree wave obeys $u(t,x) = t^{-3/2} e^{-i|x|^2/(4t)} e^{-i\\log(t)\\phi_\\infty(x/t)} h_\\infty(x/t) + O_{L^\\infty_x}(t^{-3/2-\\delta_0})$. Here $\\rho_\\infty(v)=\\int f_\\infty^2(x,v)\\,dx$, $\\phi_\\infty = V*\\rho_\\infty$, and $F_\\infty = -\\nabla(V*|h_\\infty|^2)$, so the asymptotic fermionic state sets the bosonic phase and the asymptotic bosonic state bends the fermionic trajectories. The same coupled argument, with integrable time weights, yields ordinary scattering for inverse-power interactions $V(x)=\\lambda|x|^{-\\alpha}$, $1<\\alpha<3/2$, with rates $t^{1-\\alpha}$.","pith_inferences":["A natural next step, not taken in the paper, is to construct the modified wave operator: Theorem 1.1 gives the asymptotic pair $(f_\\infty,h_\\infty)$ that a direct scattering map would need to hit, and the effective fields $F_\\infty,\\phi_\\infty$ are exactly the data such an operator would invert.","The common self-similar variable $y=x/t$ suggests the method may carry over to other kinetic–dispersive systems with borderline long-range interactions, such as Vlasov–Nordström or Vlasov–Schrödinger with gravitational coupling, where a similar two-sided log correction should appear.","The restriction $\\alpha<3/2$ is technical (an endpoint Sobolev embedding); a sharper embedding or a different norm might push the short-range result up to $\\alpha=2$, and the failure at $\\alpha=3/2$ would then be a genuine threshold rather than an artifact.","One testable consequence of the coupling is that the asymptotic phase $\\phi_\\infty$ should equal the Coulomb potential of the limiting fermionic spatial density, equation (4.24); measuring the phase difference in a numerical two-species simulation would verify the self-consistent closure of the asymptotic data."],"forward_implications":["The asymptotic state of the system is genuinely coupled: $f_\\infty$ and $h_\\infty$ cannot be read off from the free evolution of the initial data, because each carries a leading-order imprint of the other.","The fermionic sector satisfies a modified scattering law along the corrected characteristics $x+tv-\\log(t)F_\\infty(v)$, so the long-time force on the fermions is entirely described by the limiting bosonic profile.","The bosonic sector obeys $u(t,x)\\approx t^{-3/2}e^{-i|x|^2/(4t)}e^{-i\\log(t)\\phi_\\infty(x/t)}h_\\infty(x/t)$, meaning the fermionic density acts as a logarithmic phase clock on the condensate.","For $1<\\alpha<3/2$, the same bootstrap yields ordinary (unmodified) scattering with the explicit rates (1.11)–(1.12), confirming that $\\alpha=3/2$ is the borderline for long-range behavior in this coupled setting.","Conservation laws imply the constructed asymptotic profiles preserve the initial fermionic and bosonic masses, so the scattering map, once defined, will be mass-preserving."],"supporting_citations":[{"why":"It supplies the physical-space Hartree asymptotic estimates that the present proof treats as its main dispersive input.","marker":"[25]"},{"why":"It provides the explicit asymptotic dynamics for Vlasov–Poisson that the authors adapt to the kinetic component here.","marker":"[17]"},{"why":"It establishes that linear scattering is not the correct asymptotic description for Vlasov–Poisson, motivating the modified scattering framework.","marker":"[5]"},{"why":"It develops modified wave operators and large-time asymptotics for the long-range Hartree equation, the bosonic model underlying (1.1b).","marker":"[7]"},{"why":"It gives the classical asymptotic analysis for Hartree equations, which the authors cite as the baseline that cannot be directly applied to the coupled system.","marker":"[10]"},{"why":"It offers the Fourier-space proof of Hartree modified scattering, the alternative route the authors say is not available for the coupled system.","marker":"[18]"},{"why":"It established a large-data global existence theory for the Coulomb Vlasov–Hartree system, providing the prior well-posedness baseline.","marker":"[4]"}],"fun_headline_variants":["Coulomb Vlasov–Hartree: log-corrected scattering for small data","Coulomb Bose–Fermi mixture: log corrections set each species' motion","Log-corrected Coulomb scattering for Bose–Fermi mixtures","Small Coulomb data yield global Vlasov–Hartree solutions with log drift"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole long-time picture rests on the small-data bootstrap closing with the logarithmic growth rates stated in Proposition 3.10, on the validity of the unpublished companion's physical-space Hartree estimates [25] used as a black box, and on smallness in (1.7) implying the $C^1$ regularity that the local well-posedness argument requires.","fun_headline_variants_meta":{"raw":{"variants":["Coulomb Vlasov–Hartree: log-corrected scattering for small data","Coulomb Bose–Fermi mixture: log corrections set each species' motion","Log-corrected Coulomb scattering for Bose–Fermi mixtures","Small Coulomb data yield global Vlasov–Hartree solutions with log drift"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001223,"raw_usage":{"total_tokens":5019,"prompt_tokens":927,"completion_tokens":4092,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":4008}},"tokens_in":543,"tokens_out":4092,"duration_ms":36868,"temperature":1.0,"reasoning_tokens":4008,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:36:32.350040+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve (1.1) numerically in three dimensions for small Gaussian data with $V(x)=\\lambda|x|^{-1}$ and check whether $\\tilde\\gamma(t,x,v)=\\gamma(t,x-\\log(t)F_\\infty(v),v)$ converges in $L^\\infty_{x,v}$ with the stated $t^{-\\delta_0}$ rate and whether the rescaled Schrödinger profile, after removing the phase $e^{-i\\log(t)\\phi_\\infty(x/t)}$, converges in $L^\\infty_x$; failure of either for arbitrarily small data would refute Theorem 1.1. Alternatively, finding data satisfying (1.7) but not $C^1$ for which the system has no strong solution would refute the theorem's regularity statement.","supporting_citations":[{"cited_title":"Pausader and M","cited_arxiv_id":null,"evidence_quote":"It supplies the physical-space Hartree asymptotic estimates that the present proof treats as its main dispersive input."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the explicit asymptotic dynamics for Vlasov–Poisson that the authors adapt to the kinetic component here."},{"cited_title":"Choi and S","cited_arxiv_id":null,"evidence_quote":"It establishes that linear scattering is not the correct asymptotic description for Vlasov–Poisson, motivating the modified scattering framework."},{"cited_title":"Ginibre and T","cited_arxiv_id":null,"evidence_quote":"It develops modified wave operators and large-time asymptotics for the long-range Hartree equation, the bosonic model underlying (1.1b)."},{"cited_title":"Hayashi and P","cited_arxiv_id":null,"evidence_quote":"It gives the classical asymptotic analysis for Hartree equations, which the authors cite as the baseline that cannot be directly applied to the coupled system."},{"cited_title":"Kato and F","cited_arxiv_id":null,"evidence_quote":"It offers the Fourier-space proof of Hartree modified scattering, the alternative route the authors say is not available for the coupled system."},{"cited_title":"Global Existence and Time Decay for the Vlasov-Hartree System","cited_arxiv_id":"2607.04444","evidence_quote":"It established a large-data global existence theory for the Coulomb Vlasov–Hartree system, providing the prior well-posedness baseline."}],"review_version":1}