{"id":"283376c7-c7a1-4109-a0cc-ae1c17c0cd21","arxiv_id":"2506.07215","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For small H4-perturbations of the 3D compressible viscoelastic system in an exterior domain, the paper proves L2 decay at rate t^{-3/4}, derivatives at t^{-5/4}, and L∞ decay at t^{-3/2}, though the proof requires an L1 assumption not stated in the theorem.","lead":"This paper analyzes how small disturbances of a compressible elastic fluid in the space outside an obstacle decay over time, proving rates of t^{-3/4} in L2 and faster for derivatives. It is the first decay result for this exterior-domain problem, although the main theorem as stated omits an L1 condition that the proof silently uses.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1's hypotheses omit an L1 condition the proof needs: displays (6.7) and (6.16) invoke Theorem 2.2(A) with q=1, which requires U0∈L1, but H^4 smallness on an exterior domain does not imply L1.","rationale":"I read the paper as a serious attempt to show optimal decay for the compressible viscoelastic system in an exterior domain by combining Kobayashi–Shibata cutoff with the companion local-energy-decay result. The central theorem must follow from its stated hypotheses, and the first point where that fails is the very first estimate in the nonlinear bootstrap: Theorem 2.2 is an Lq-based linear decay result, and Section 6 selects q=1. There is no bridge from H^4 to L1 on an exterior domain, so the stated theorem is strictly stronger than what the proof establishes. This is more decisive than the unverified Lemma 3.2, because even granting that lemma the argument still has an unstated L1 hypothesis. The repair is natural—add U0∈L1 to Theorem 2.1—so the rejection is not a claim of dishonesty or of a fundamentally wrong method; it is a precise statement that the theorem as written is not proved. For that reason the reader's REJECT verdict stands unchanged. I partially agree with the reader: the L1 gap is exactly the load-bearing weakness, though I would not rank the companion-preprint dependency as the primary failure.","tokens_in":30222,"tokens_out":12833,"duration_ms":140413,"concrete_test":"Check the proof by attempting to derive (6.7) from the stated hypotheses without any L1 bound: take the test data (v0=0, E0=∇ψ, n0=−Δψ with ψ=ε/r for large r in the exterior domain) and verify that it satisfies the hypotheses literally stated in Theorem 2.1 while ∥U0∥1=∞; then confirm that every application of Theorem 2.2 in Step 1 and Step 2 is at q=1 with an explicit ∥U0∥1 term. If (6.7) cannot be applied to this data, the proof gap is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 6 opens its bootstrap at display (6.7) with ∥e^{tA}U0∥2 ≤ C t^{-3/4}(∥U0∥1+∥U0∥_{W^{1,0}_2}) and ∂x/∂t variants, i.e. Theorem 2.2(A)/(B) at q=1; display (6.16) uses the same q=1 for the L∞ bound. The stated hypotheses of Theorem 2.1 are second-order compatibility/regularity, ∇ρ0+divF0^T=0, and small ∥(ρ0−1,u0,F0−I)∥_{H^4}. The compatibility condition defined in §1 is purely a regularity condition, and neither it nor H^4 smallness implies L1 on an exterior domain. Explicitly, for Ω={|x|>1}, take a smooth ψ with ψ=ε/r for large r and ψ=0 near ∂Ω, and set v0=0, E0=∇ψ, n0=−Δψ; then ∇n0+divE0^T=0 and the H^4 norm can be made arbitrarily small, but E0∼εr^{-2}, so U0∉L1. Hence Theorem 2.2(A) with q=1 cannot be invoked, and the rates (6.7), (6.16), and all later M2, M∞, M2 bounds are unsupported. This holds even if Lemma 3.2 is accepted; it is an internal gap in the nonlinear argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the three-dimensional compressible viscoelastic equations posed in an exterior domain with no-slip boundary condition. The main result, Theorem 2.1, claims that under a second-order compatibility/regularity condition, the linearized structural identity ∇ρ0+divF0^T=0, and smallness of the H^4 norm of the perturbation, the solution decays as ∥(ρ-1,u,F-I)(t)∥_2=O(t^{-3/4}), the first spatial and temporal derivatives decay as O(t^{-5/4}), and the L∞ norm decays as O(t^{-3/2}). The proof proceeds by establishing linear semigroup estimates (Theorem 2.2) via a cut-off argument that combines local energy decay in the exterior domain with Fourier/spectral estimates in R^3, and then by a Duhamel bootstrap for the nonlinear problem in Section 6.","tokens_in":30481,"tokens_out":10028,"duration_ms":98903,"significance":"If the main theorem were valid, it would provide the first optimal decay rates for the compressible viscoelastic system in an exterior domain, extending known Cauchy-problem results of Hu-Wu and others to the boundary-value setting. The paper contains explicit decay rates, a transparent spectral analysis in R^3 following the Hu-Wu decomposition, and no fitted parameters; the claimed rates are falsifiable and in line with the expected diffusion-wave behavior. However, the central theorem is not established by the proof as written: two hypotheses required by the linear estimates are absent from Theorem 2.1, and a key local energy decay lemma is imported from an unreviewed companion preprint. These issues affect the foundation of the paper, so the significance is conditional and the current manuscript cannot be accepted.","major_comments":[{"comment":"The proof of Theorem 2.1 invokes Theorem 2.2(A) with q=1, which requires U0∈L1, but the hypotheses of Theorem 2.1 (second-order compatibility/regularity, ∇ρ0+divF0^T=0, and small H^4 norm) do not imply U0∈L1 on an exterior domain. For Ω={|x|>1}, take a smooth ψ with ψ=ε/r for large r and ψ=0 near ∂Ω, and set v0=0, E0=∇ψ, n0=−Δψ. Then ∇n0+divE0^T=0, the H^4 norm can be made arbitrarily small by choosing ε small, but E0∼εr^{-2}, so U0∉L1. Consequently the q=1 estimates in (6.7) and (6.16), and all subsequent bounds for M2(t), M∞(t), and the Step 3 quantity M2(t), are unsupported. The theorem as stated is therefore not proved.","section":"Section 6, Eqs. (6.7) and (6.16)"},{"comment":"Theorem 2.1 omits the second structural condition in (1.3), namely F0^{lk}∂_lF0^{ij}=F0^{lj}∂_lF0^{ik} (or its linearized form ∂_{x_l}E0_{jk}=∂_{x_k}E0_{jl}). The linear estimates in Section 4, specifically the derivation of (4.4)-(4.6) and the reduction of the elastic part to a curl system, use this condition; the R^3 estimates of Theorem 4.1 and hence Theorem 2.2 depend on it. The '2nd order compatibility condition' defined in Section 1 is only a regularity condition and does not include this identity. Without adding the structural assumption, the linear decay estimates do not apply to the class of initial data considered in Theorem 2.1.","section":"Section 2, Theorem 2.1; Section 4, Eqs. (4.4)-(4.6)"},{"comment":"The local energy decay estimate of Lemma 3.2 is quoted from the companion preprint [35] and is not proved in this manuscript. This lemma is the foundation of the cut-off argument in Section 5 and therefore of Theorem 2.2; if the estimate in [35] is incorrect or unavailable, the main theorem collapses. The authors should either provide a proof of Lemma 3.2 or state explicitly that the main result is conditional on an unreviewed preprint. As it stands, a load-bearing part of the proof is outside the manuscript.","section":"Section 3, Lemma 3.2"}],"minor_comments":[{"comment":"The introduction mentions a 'positive parameter α' representing the speed of shear waves, but no α appears in the system (1.1); this appears to be a leftover from an earlier version and should be removed or defined.","section":"Section 1"},{"comment":"The phrase 'Then, we consider t≤2' after already treating 0≤t≤2 appears to be a typo for 't≥2'.","section":"Section 6, Step 1"},{"comment":"The symbol M2(t) is defined twice with different norms: once in Step 1 and again in Step 3. This overloaded notation makes the bootstrap argument harder to follow; distinct names, such as M2^{(1)}(t) and M2^{(2)}(t), should be used.","section":"Throughout"},{"comment":"There are numerous typos and grammar issues, e.g., 'hydrodnamics', 'diff-isomorphisms', 'wriiten', 'coeffcients', 'asme', and 'provides that'. A careful proofreading is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper has a clear structure and the intended strategy is plausible, but the main theorem as stated is not supported by the proof. The missing L1 hypothesis and the missing curl-free structural condition are not cosmetic: they are used essentially in the linear and nonlinear arguments, and the first cannot be recovered from H^4 smallness on an exterior domain. In addition, the reliance on Lemma 3.2 from an unpublished companion preprint makes the paper non-self-contained at a central point. These issues could in principle be repaired by adding the missing hypotheses and waiting for [35] to be published, but that changes the statement of Theorem 2.1 substantially. I therefore recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Know this one mainly for its clean identification of an open problem: exterior-domain decay for compressible viscoelasticity, previously done only on Cauchy and bounded domains. The authors follow the Kobayashi–Shibata cutoff strategy and isolate the linear decay estimate as the key input, with the nonlinear Duhamel bootstrap handled in a fairly standard way. Section 5's cut-off argument is careful and the rates they get match the optimal whole-space rates. That part reads seriously.\n\nThe catch is that Theorem 2.1 as stated is not proved. The proof in Section 6 repeatedly uses the q=1 case of Theorem 2.2, which requires U0 in L1. See (6.7) and (6.16). The hypotheses of Theorem 2.1 give only H^4 smallness, second-order compatibility, and the linearized divergence constraint. H^4 smallness on an exterior domain does not imply L1—take ψ=ε/r for large r, E0=∇ψ, n0=−Δψ, v0=0; the H^4 norm is small, but U0 is not integrable. So the bootstrap's starting estimates are unsupported. This is not a minor technicality: the decay rates stated in the abstract and Theorem 2.1 depend on that L1 input. The right fix is probably to add an L1 (or suitable Besov) assumption to the theorem, or to work with a weighted norm from the start. The authors might also want to state explicitly that they are using the companion preprint Ishigaki–Kobayashi [35] for the local energy decay estimate; if that estimate fails, Theorem 2.2 and the whole nonlinear argument collapse. This dependency is external but the paper does disclose it, and it is a plausible result.\n\nThere are also smaller blemishes: typos, an unused α in the introduction, and a missing 'Proposition Ap.2' in the appendix. None of these affect the main gap.\n\nSo: worth a referee's time, but not acceptable as is. I would send it out, with the expectation that the L1 question is resolved and the reliance on the preprint is cleared up. If the authors add the missing assumption, the result is likely a genuine first in this setting.","headline":"The main theorem is missing an L1 hypothesis the proof actually uses; the paper is serious and the fix is likely straightforward, but as written the central decay result is unsupported.","tokens_in":31078,"tokens_out":4817,"would_cite":false,"duration_ms":45731,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76A10","35Q35","35G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the compressible viscoelastic equations in an exterior domain have strong solutions whose perturbations decay at the rates O(t^{-3/4}) in L2, O(t^{-5/4}) for first spatial/time derivatives, and O(t^{-3/2}) in L∞.","keywords":["compressible viscoelastic equations","exterior domain","decay rates","L2 estimates","linearized semigroup","local energy decay","strong solutions"],"falsifier":"For a fixed exterior domain such as the complement of a ball, with data concentrated near the boundary, compute the linearized semigroup numerically; if ||$e^{{tA}}$U0||_{$W^{{1,2}}$_2(Ω_b)} decays slower than $t^{{-2}}$ for compactly supported U0, Lemma 3.2 is false and the stated rates fail. Alternatively, exhibit small H4 data satisfying the compatibility and structural conditions but with a non-integrable algebraic tail, e.g., |U0(x)| ~ |x|^{-2}, whose solution decays slower than $t^{{-3/4}}$ in L2; that would refute Theorem 2.1 as stated.","tokens_in":29952,"feed_emoji":"🌊","tokens_out":6740,"duration_ms":63379,"temperature":0.7,"pith_summary":"The paper establishes that the three-dimensional compressible viscoelastic equations around a fixed obstacle return to equilibrium at specific polynomial rates, provided the initial perturbation is small in H4 and obeys a structural condition. It proves the solution itself decays at rate $t^{{-3/4}}$ in L2, its first spatial and time derivatives at $t^{{-5/4}}$, and its sup norm at $t^{{-3/2}}$, matching rates previously known for the whole-space problem. The proof proceeds by first deriving L2 decay estimates for the linearized system, combining local energy decay near the obstacle with whole-space spectral estimates, and then closing the nonlinear argument through Duhamel's formula.","feed_headline":"Viscoelastic flows around obstacles decay at optimal rates","feed_subtitle":"Proof gives t^{-3/4} L2 decay for the perturbation and faster rates for derivatives in exterior domains.","key_machinery":"The object that carries the argument is the semigroup $e^{{tA}}$ generated by the linearization of the equations around (1,0,I), with A acting on U=(n,v,E). The mechanism is a two-region decomposition: near the obstacle the solution is controlled by a local energy decay estimate giving ||∂t^m $e^{{tA}}$U0||_{$W^{{1,2}}$_2(Ω_b)} ≤ C $t^{{-2-m}}$ ||U0||_{$W^{{1,0}}$_2}, while away from the obstacle the solution is controlled by whole-space estimates obtained from the Helmholtz decomposition v=-$Λ^{{-1}}$∇d-$Λ^{{-1}}$divω, which reduces the linear system to independent Fourier-analyzed 2×2 systems whose eigenvalue expansions produce the $t^{{-3/(2q)}}$ decay rates. The two pieces are stitched together with cut-off functions and Duhamel's formula.","core_discovery":"The central discovery is that, for the compressible viscoelastic system in the exterior of a bounded domain, solutions starting close to the equilibrium state (ρ,u,F)=(1,0,I) in H4, satisfying the second-order compatibility condition, and obeying the structural constraint ∇ρ0+divF0^T=0, decay at the optimal rates ||(ρ-1,u,F-I)(t)||_2=O($t^{{-3/4}}$), ||∂x(ρ,u,F)(t)||_{$W^{{1,2}}$_2}+||∂t(ρ,u,F)(t)||_{1,2}=O($t^{{-5/4}}$), and ||(ρ-1,u,F-I)(t)||_∞=O($t^{{-3/2}}$). The proof works through the linearized semigroup $e^{{tA}}$: a cut-off splits the solution into a compactly supported part governed by local energy decay near the obstacle and a far-field part whose decay comes from Fourier analysis of the coupled systems for the compressible and incompressible parts of the velocity.","pith_inferences":["The proof as written invokes Theorem 2.2 with q=1 in Section 6, which requires U0∈L1, yet Theorem 2.1 states only H4 smallness; if L1 integrability is genuinely needed, the theorem's statement is incomplete and the rates may fail for data with slow algebraic tails.","The local energy decay estimate quoted from the companion preprint is the single unproved input; any weakening of its t^{-2-m} decay would directly lower the final rates.","The same cut-off plus spectral-split strategy should apply to other dissipative systems with a diffusion-wave structure in exterior domains, such as compressible Navier-Stokes or magnetohydrodynamics, giving analogous optimal rates.","A numerical study of the semigroup for a simple obstacle, such as a ball, could test the local energy decay exponent directly and would also reveal whether the L1 assumption is avoidable."],"forward_implications":["If Theorem 2.1 is correct, the critical decay rates known for the Cauchy problem on R^3 are recovered in exterior domains, so the boundary does not slow the asymptotic return to equilibrium.","The rates O(t^{-3/4}) for the L2 norm and O(t^{-3/2}) for the L∞ norm match the linear diffusion-wave structure, and the nonlinear iteration closes at exactly these orders.","The second-order compatibility condition and smallness of the H4 norm are sufficient for these rates, with higher regularity presumably giving correspondingly faster higher-derivative decay.","The structural constraint ∇ρ0+divF0^T=0, the linearized form of div(ρF^T)=0, keeps the linear system in the 2×2 split form; removing it would change the spectral structure and likely the rates."],"supporting_citations":[{"why":"Supplies the local energy decay estimate (Lemma 3.2) on which the near-field decay in Theorem 2.2 is built.","marker":"[35]"},{"why":"Provides the cut-off technique and L2 decay estimates for semigroups in exterior domains that organize Section 5.","marker":"[21]"},{"why":"Supplies the Helmholtz decomposition and spectral simplification for the whole-space linearized system used in Section 4.","marker":"[14]"},{"why":"Establishes that the constraint div(ρF^T)=0 is preserved by the flow, used to reduce the nonlinear system.","marker":"[12]"},{"why":"Provides the analytic semigroup and resolvent estimates for the linearized operator stated in Lemma 3.1.","marker":"[20]"}],"fun_headline_variants":["Optimal t^-3/4 decay for viscoelastic exterior flows","Viscoelastic decay outside obstacles hits optimal rate","Exterior viscoelastic flows decay at optimal rates","Optimal L2 decay proven for viscoelastic exterior flows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument collapses if the quoted local energy decay estimate (Lemma 3.2, taken from a companion preprint) is false, and it also silently uses L1 integrability of the initial perturbation in Section 6 even though Theorem 2.1 does not state that assumption.","fun_headline_variants_meta":{"raw":{"variants":["Optimal t^-3/4 decay for viscoelastic exterior flows","Viscoelastic decay outside obstacles hits optimal rate","Exterior viscoelastic flows decay at optimal rates","Optimal L2 decay proven for viscoelastic exterior flows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001638,"raw_usage":{"total_tokens":6443,"prompt_tokens":810,"completion_tokens":5633,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":426,"completion_tokens_details":{"reasoning_tokens":5565}},"tokens_in":426,"tokens_out":5633,"duration_ms":37761,"temperature":1.0,"reasoning_tokens":5565,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:39:24.567182+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed exterior domain such as the complement of a ball, with data concentrated near the boundary, compute the linearized semigroup numerically; if ||$e^{{tA}}$U0||_{$W^{{1,2}}$_2(Ω_b)} decays slower than $t^{{-2}}$ for compactly supported U0, Lemma 3.2 is false and the stated rates fail. Alternatively, exhibit small H4 data satisfying the compatibility and structural conditions but with a non-integrable algebraic tail, e.g., |U0(x)| ~ |x|^{-2}, whose solution decays slower than $t^{{-3/4}}$ in L2; that would refute Theorem 2.1 as stated.","supporting_citations":[{"cited_title":"Local energy decay of solutions to the linearized compressible viscoelastic system around motionless state in an exterior domain","cited_arxiv_id":"2505.23169","evidence_quote":"Supplies the local energy decay estimate (Lemma 3.2) on which the near-field decay in Theorem 2.2 is built."},{"cited_title":"Kobayashi and Y","cited_arxiv_id":null,"evidence_quote":"Provides the cut-off technique and L2 decay estimates for semigroups in exterior domains that organize Section 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Helmholtz decomposition and spectral simplification for the whole-space linearized system used in Section 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that the constraint div(ρF^T)=0 is preserved by the flow, used to reduce the nonlinear system."},{"cited_title":"Kobayashi","cited_arxiv_id":null,"evidence_quote":"Provides the analytic semigroup and resolvent estimates for the linearized operator stated in Lemma 3.1."}],"review_version":1}