{"id":"2f02defe-47a8-427d-82dc-3eb00ec0dbc3","arxiv_id":"1908.04080","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves global-in-time L^q-L^r decay estimates for the gradient of a generalized Oseen evolution operator in 3D exterior domains, including the optimal rate for q≤r≤3 and an L^q-L^∞ bound, for bounded Hölder-continuous time-dependent translation and rotation.","lead":"This mathematics paper proves optimal decay rates for the spatial gradient of fluid motion induced by a moving, rotating body, in the linearized setting. It settles an open problem and gives tools for proving stability of Navier-Stokes flows around rigid bodies with time-dependent motion.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof of Theorem 2.1 is consistent and the compressed interpolation step for r>3 is standard.","rationale":"The reader's ACCEPT verdict is justified. I examined the local energy decay propositions, the parametrix construction, the pressure estimates, and the final cut-off argument. The proof is long but the dependencies on [33] and [26] are transparent, and the novel steps are detailed. The only place requiring the reader to fill in a nontrivial but standard argument is the interpolation to r>3; this is not a correctness risk. The bounded-Hölder assumption on (η,ω) is a stated, natural restriction rather than a hidden one. No significant mathematical objection surfaced, so the verdict should remain unchanged.","tokens_in":35209,"tokens_out":29414,"duration_ms":281099,"concrete_test":"Recompute the r>3 case of Theorem 2.1 by writing ||∇T(t,s)f||_{L^r} ≤ C ||∇T(t,s)f||_{L^3}^{3/r} ||T(t,s)f||_{L^∞}^{1-3/r} (with L^q in place of L^3 when q>3), using (2.22) with r=3 and (2.24) to check that the resulting exponent is exactly -3/(2q) for every r>3; if the exponents match, the compressed step is fully validated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the main chain: Theorem 2.1 is reduced to the local energy decay estimates (Propositions 6.1 and 6.2) and the outer decay estimates (7.1)-(7.2), via the whole-space Duhamel formula (7.4). The pressure and time-derivative terms are controlled by Corollary 6.1, and the weight condition q>3/2 in Lemma 5.2 is correctly stated. The weakest-looking spot is the final passage from (7.1)-(7.2) to (2.23) for r>3, which the paper does not spell out; however, it follows by a standard Gagliardo-Nirenberg interpolation between the L^q or L^3 gradient bound (rate t^{-3/(2q)}) and the L^∞ bound (2.24) for T(t,s) (also rate t^{-3/(2q)}). I therefore found no load-bearing flaw. The bounded-Hölder assumption (1.2) is explicit and natural, and the dependence of constants on the bound m is consistent with the prior results from [33] used as ingredients.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves global-in-time L^q-L^r decay estimates for the gradient of the evolution operator T(t,s) associated with the linearized non-autonomous Oseen system (1.1) in a three-dimensional exterior domain, under bounded and θ-Hölder continuous translational and angular velocities satisfying (1.2). Theorem 2.1 establishes the optimal rate (2.22) for 13, and the L^q-L^∞ estimate (2.24). Theorem 2.2 provides a Lorentz-space version for the adjoint operator and derives the integrable estimate (2.26). The proof combines the author's earlier L^q-L^r estimates for T(t,s) from [33] with new local energy decay estimates (Propositions 6.1 and 6.2) obtained without spectral analysis, and with whole-space Duhamel estimates near spatial infinity. The paper also identifies and corrects an oversight in Hansel and Rhandi [26] concerning the regularity space Z_q(D).","tokens_in":35410,"tokens_out":25537,"duration_ms":238835,"significance":"If correct, the result is a significant unified extension of the autonomous Stokes, Oseen, and rotating-obstacle semigroup estimates: it recovers the known three-dimensional L^q-L^r gradient decay rates and covers time-dependent rigid motions with bounded Hölder velocities. The proof is technically detailed, with explicit dependence of the constants on the bound m in (2.21), and the treatment of ∂tT in W^{-1,q} is a genuinely new ingredient. The correction of the Z_q(D) regularity gap in [26] is a valuable contribution. The main limitation—bounded, Hölder continuous body velocities—is stated clearly, and the remaining difficulties in 2D and at endpoint cases are honestly acknowledged. The paper contains no machine-checked proofs or code, but the analytic arguments are structured, checkable, and the main theorem is precisely stated.","major_comments":[],"minor_comments":[{"comment":"The passage from (7.1)–(7.2) to the reduced rate (2.23) for r>3 is compressed to a reference to the semigroup property; I recommend spelling out the midpoint split T(t,s)=T(t,(t+s)/2)T((t+s)/2,s), where the first factor is estimated by (2.20) from L^q to L^r and the second by (7.1) with index r, whose decay rate 3/(2r) cancels the r-dependence and yields the claimed exponent -3/(2q).","section":"Section 7, proof of Theorem 2.1"},{"comment":"In the reduction of the time-derivative estimate to q ∈ (3/2,∞), the application of (5.24) with q=3 requires T((t+s)/2,s)f ∈ Z_3(D); since the proof has already reduced to f ∈ C∞_{0,σ}(D), this regularity follows from Proposition 2.1, but the step should be stated explicitly for the reader.","section":"Section 6, proof of Proposition 6.2"},{"comment":"The statement of (2.23) is restricted to t-s>2; for t-s≤2 the same bound is implied by Proposition 2.2(2) because the exponent in (2.22) is more negative than -3/(2q) when r>3, so a remark noting that (2.23) actually holds for all t>s≥0 would avoid a misleading restriction.","section":"Theorem 2.1(2)"},{"comment":"The manuscript contains numerous typographical errors that should be corrected, including 'matirices' (page 2), 'thst' (page 15), 'decompostion' (page 8), 'associsted' (page 20), 'argumant' (page 32), 'infinifity' (page 4), and 'worse' where 'worth' is intended (page 13).","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"To the editor: This is a serious, technically demanding contribution. The author's reliance on his own earlier paper [33] is legitimate and fully disclosed; the new gradient estimates are not circular. The compressed passages noted in the report are presentation issues rather than correctness concerns. The paper fits the scope of the journal and, after the requested clarifications, should be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a good paper, worth sending out. The main result is what it says: for the linearized non-autonomous system in a 3D exterior domain, with time-dependent translational and angular velocities satisfying bounded Hölder continuity, the gradient of the evolution operator obeys optimal L^q-L^r decay rates, plus an L^q-L^∞ bound. The case 13 gets the reduced t^{-3/(2q)}; the L∞ bound on T(t,s) itself is new. The gradient question was explicitly left open in the author's previous paper [33], so this finishes the program.\n\nWhat's genuinely new: the local energy decay estimates and the handling of ∂tT(t,s) in W^{-1,q} without spectral analysis. In the autonomous case, that is where the resolvent analysis lives; here the author substitutes regularity of the time derivative together with the previously established L^q-L^r estimates. He also corrects an oversight in [26] concerning the space Z_q(D) and the boundary condition for PK(t,s)f. That kind of honest repair deserves credit.\n\nSoft spots: the paper leans on the author's own [33] for the L^q-L^r estimates of T(t,s) and the adjoint. That is transparent and not circular—the old theorem did not assume gradient decay—but it does mean the reader needs the 2018 paper to verify the input. The r>3 passage is not fully spelled out in the text; the stress test confirms it is a standard Gagliardo-Nirenberg interpolation between the L^3 gradient bound and the L∞ bound. I would call it a minor exposition issue, not a gap. The constants depend on the sup-norm and Hölder constant of η,ω; that is explicit and natural.\n\nThe one thing I would like in a revision is a short paragraph in Section 7 laying out that interpolation step for r>3 instead of leaving it to the reader. Also, the abstract says 'completely recovers' the autonomous case; that is true for the rates, but only in 3D. The paper states that clearly enough.\n\nWho is it for: people working on fluid-structure interaction for Navier-Stokes past moving bodies, especially non-autonomous motions, and anyone needing unified gradient decay estimates for stability or attainability applications. It deserves a serious referee—this is the kind of paper that should go to peer review, not be desk-rejected.","headline":"Solid continuation that closes the open gradient decay question for the non-autonomous Oseen evolution operator, with a clean proof idea and no load-bearing flaws.","tokens_in":35906,"tokens_out":2038,"would_cite":true,"duration_ms":21401,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35B40","47D06","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves optimal $L^q$-$L^r$ decay estimates for the gradient of the evolution operator of the linearized Navier-Stokes system in the exterior of a rigid body moving with time-dependent velocities, recovering the autonomous rates.","keywords":["Oseen evolution operator","non-autonomous linearized Navier-Stokes","exterior domain","L^q-L^r decay estimates","rotating rigid body","time-dependent rigid motion","gradient estimates","local energy decay"],"falsifier":"Take a bounded periodic motion such as $\\eta(t)=\\varepsilon\\sin t$, $\\omega=0$, and measure the quantity $\\sup_{t>s,\\,t-s\\ge1}(t-s)^{1/2}\\|\\nabla T(t,s)\\|_{L^3\\to L^3}$; if this quantity is unbounded as $\\varepsilon$ grows while (2.21) holds uniformly, the claimed uniform constant in Theorem 2.1(1) fails.","tokens_in":35015,"feed_emoji":"🌊","tokens_out":9369,"duration_ms":88874,"temperature":0.7,"pith_summary":"The paper establishes global-in-time decay estimates for the gradient of the solution operator $T(t,s)$ of the linearized Navier-Stokes system in the three-dimensional exterior of a rigid body whose translational and angular velocities vary with time. The estimates recover, for the non-autonomous problem, the optimal $L^q$-$L^r$ rates known for the autonomous Stokes and Oseen semigroups: the full smoothing rate for $13$, and an $L^q$-$L^\\infty$ bound. Because the drift and Coriolis coefficients are time-dependent and unbounded in space, classical resolvent analysis near the spectral origin is unavailable; the proof replaces it with local energy decay near the obstacle and a sharp analysis of the temporal derivative of the evolution operator. These estimates are the missing piece for studying stability and attainability of Navier-Stokes flows around moving and rotating bodies.","feed_headline":"Optimal decay proven for fluid past a moving body","feed_subtitle":"New gradient estimates match the autonomous Stokes and Oseen decay rates for time-dependent rigid motions.","key_machinery":"The central object is the non-autonomous evolution operator $T(t,s)$ acting on the solenoidal Lebesgue spaces $L^q_\\sigma(D)$ over the exterior domain. The argument is carried by a parametrix that patches an explicit whole-space solution $U(t,s)$, obtained from the heat semigroup by a time-dependent change of variables, with a bounded-domain evolution operator $V(t,s)$; cut-off functions and a divergence-correction operator enforce the boundary condition. The new ingredient is the analysis of the temporal derivative $\\partial_t T(t,s)f$ in the negative Sobolev space $W^{-1,q}(D_R)$, together with local energy decay estimates near the obstacle and weighted estimates for the Helmholtz projection; this substitutes for the resolvent analysis used in the autonomous theory.","core_discovery":"Theorem 2.1 asserts that for the non-autonomous system (1.1) in a three-dimensional exterior domain, under the boundedness and Hölder-continuity assumption (1.2), the gradient of the evolution operator satisfies the optimal estimate $\\|\\nabla T(t,s)f\\|_r \\le C(t-s)^{-(3/q-3/r)/2-1/2}\\|f\\|_q$ for all $1s\\ge0$, the reduced estimate $\\|\\nabla T(t,s)f\\|_r \\le C(t-s)^{-3/(2q)}\\|f\\|_q$ for $r>3$ and $t-s>2$, and the $L^q$-$L^\\infty$ estimate with rate $(t-s)^{-3/(2q)}$; the same bounds hold for the adjoint $T(t,s)^*$. The paper states that this completely recovers the $L^q$-$L^r$ estimates for the autonomous case, including the Stokes semigroup, the Oseen semigroup, and the semigroups with constant rotation. The rates are optimal in general: earlier results for the Stokes case show that the decay cannot be improved.","pith_inferences":["A testable extension would be to relax the boundedness assumption (1.2) to local Hölder regularity together with some time-averaged integrability; if the constants depend only on sliding-window norms, the same decay rates should persist for motions that accelerate over finite time horizons.","The local-energy-decay route may transfer to two-dimensional exterior domains, where the analogous gradient decay for rotating obstacles has remained open.","The adjoint estimate (2.26) suggests, though the paper does not state it, that time-periodic background flows with zero-mean translation should be nonlinearly stable at the scale-critical weak-$L^3$ level.","The conjecture mentioned in Remark 2.1, that the full smoothing rate could hold for $q\\le r\\le 6$ when translation is present, could be probed by studying the singularity of the Oseen resolvent near the spectral origin."],"forward_implications":["The gradient estimate for $q=r=3$ upgrades previously known global-in-time constructions of Navier-Stokes flows around moving bodies to include the large-time behavior of the velocity gradient.","The theorem unifies the known $L^q$-$L^r$ decay theory for the autonomous Stokes, Oseen, and rotating-body semigroups in three-dimensional exterior domains.","The adjoint Lorentz-space estimate (2.25) with $r=3$ yields the space-time integrability (2.26), the standard input for stability and attainability of background flows with scale-critical far-field decay.","The non-autonomous estimates make the starting problem for a body rotating from rest accessible, since they avoid the unbounded-coefficient obstruction that blocked the earlier autonomous approach.","The reduced rate $(t-s)^{-3/(2q)}$ for $r>3$ is optimal in the Stokes case, while the possibility of a better rate when translation is present remains open."],"supporting_citations":[{"why":"Constructs the evolution operator $T(t,s)$ and supplies the initial-time smoothing estimates from which the global decay argument starts.","marker":"[26]"},{"why":"Establishes the global-in-time $L^q$-$L^r$ estimates of $T(t,s)$ and its adjoint that the present gradient estimates extend.","marker":"[33]"},{"why":"Provides the Stokes-semigroup baseline showing the claimed gradient decay rates are optimal.","marker":"[42]"},{"why":"Supplies the Oseen-semigroup $L^q$-$L^r$ estimates that Theorem 2.1 recovers as the autonomous translation-only case.","marker":"[39]"},{"why":"Gives the autonomous rotating-body semigroup estimates and the pressure estimate that motivate the non-autonomous analysis.","marker":"[36]"},{"why":"Treats the autonomous Oseen semigroup with both translation and rotation, another case recovered by the unified theorem.","marker":"[44]"},{"why":"Provides the interpolation argument that turns the adjoint Lorentz-space gradient bounds into the integrability estimate (2.26).","marker":"[52]"},{"why":"Supplies the iterated-convolution lemma used to sum the parametrix series for the evolution operator.","marker":"[21]"},{"why":"Provides the weighted $L^q$ estimates for singular integrals needed to control the Helmholtz projection in the auxiliary spaces.","marker":"[12]"},{"why":"Defines the divergence-correction operator used in the cut-off procedure to maintain the solenoidal condition.","marker":"[2]"}],"fun_headline_variants":["Optimal decay for fluid past time-dependent rotating body","Gradient decay matches autonomous Stokes/Oseen rates","Sharp gradient bounds for non-autonomous Oseen operator","Optimal Lq-Lr gradient decay for time-dependent rigid motion","Exterior flow decay sharpened to optimal gradient rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The estimates require the prescribed translational and angular velocities to be bounded and Hölder-continuous in time; if a body speeds up or spins faster without bound, the global-in-time constants need not stay finite.","fun_headline_variants_meta":{"raw":{"variants":["Optimal decay for fluid past time-dependent rotating body","Gradient decay matches autonomous Stokes/Oseen rates","Sharp gradient bounds for non-autonomous Oseen operator","Optimal Lq-Lr gradient decay for time-dependent rigid motion","Exterior flow decay sharpened to optimal gradient rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000612,"raw_usage":{"total_tokens":2907,"prompt_tokens":1066,"completion_tokens":1841,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":682,"completion_tokens_details":{"reasoning_tokens":1763}},"tokens_in":682,"tokens_out":1841,"duration_ms":13805,"temperature":1.0,"reasoning_tokens":1763,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:52:36.170892+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a bounded periodic motion such as $\\eta(t)=\\varepsilon\\sin t$, $\\omega=0$, and measure the quantity $\\sup_{t>s,\\,t-s\\ge1}(t-s)^{1/2}\\|\\nabla T(t,s)\\|_{L^3\\to L^3}$; if this quantity is unbounded as $\\varepsilon$ grows while (2.21) holds uniformly, the claimed uniform constant in Theorem 2.1(1) fails.","supporting_citations":[{"cited_title":"Hansel and A","cited_arxiv_id":null,"evidence_quote":"Constructs the evolution operator $T(t,s)$ and supplies the initial-time smoothing estimates from which the global decay argument starts."},{"cited_title":"Hishida, Large time behavior of a generalized Oseen evolution o perator, with applications to the Navier-Stokes flow past a rotating obstacle, Math","cited_arxiv_id":null,"evidence_quote":"Establishes the global-in-time $L^q$-$L^r$ estimates of $T(t,s)$ and its adjoint that the present gradient estimates extend."},{"cited_title":"Maremonti and V.A","cited_arxiv_id":null,"evidence_quote":"Provides the Stokes-semigroup baseline showing the claimed gradient decay rates are optimal."},{"cited_title":"Kobayashi and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the Oseen-semigroup $L^q$-$L^r$ estimates that Theorem 2.1 recovers as the autonomous translation-only case."},{"cited_title":"Hishida and Y","cited_arxiv_id":null,"evidence_quote":"Gives the autonomous rotating-body semigroup estimates and the pressure estimate that motivate the non-autonomous analysis."},{"cited_title":"Shibata, On the Oseen semigroup with rotating effect, Functional Analysis and Evolution Equations, The G¨ unter Lumer Volume, 595–611, Birkh¨ auser, Basel, 2008","cited_arxiv_id":null,"evidence_quote":"Treats the autonomous Oseen semigroup with both translation and rotation, another case recovered by the unified theorem."},{"cited_title":"Yamazaki, The Navier-Stokes equations in the weak- Ln space with time- dependent external force, Math","cited_arxiv_id":null,"evidence_quote":"Provides the interpolation argument that turns the adjoint Lorentz-space gradient bounds into the integrability estimate (2.26)."},{"cited_title":"Geissert, H","cited_arxiv_id":null,"evidence_quote":"Supplies the iterated-convolution lemma used to sum the parametrix series for the evolution operator."},{"cited_title":"Farwig and H","cited_arxiv_id":null,"evidence_quote":"Provides the weighted $L^q$ estimates for singular integrals needed to control the Helmholtz projection in the auxiliary spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the divergence-correction operator used in the cut-off procedure to maintain the solenoidal condition."}],"review_version":1}