{"id":"a2b13ac3-33b4-466d-95e1-d0d70f2e92ee","arxiv_id":"2506.22155","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An energy estimate and existence of weak solutions are proved for the 3D heat-conducting Navier-Stokes system in a cylinder with arbitrarily large inflow and outflow.","lead":"The paper proves an energy estimate and existence of weak solutions for a heat-conducting incompressible fluid in a cylinder with large inflow and outflow, requiring no smallness assumptions on the data. It extends earlier existence results by deriving two-sided temperature bounds, which relaxes restrictions on the temperature-dependent force coefficient.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The energy estimate in Lemma 3.2 introduces an unproved L3 forcing term in (3.32) and absorbs it using an L6/5 threshold; under the stated assumptions the absorption fails, so the Main Theorem is not established as written.","rationale":"The reader's rationale already flags the (3.38) absorption as the main gap, and I agree that this is the most load-bearing issue: it breaks the proof of Lemma 3.2 and propagates into the Galerkin compactness estimate (4.5), so the Main Theorem is not proven under the stated assumptions. I therefore keep the conditional verdict rather than escalating: the gap is concrete and likely fixable by adding the missing L^3 assumption on ωf, or by replacing the ϑ-dependent force estimate with one that uses only L^{6/5}. The reader's formal weakest_assumption, however, is the weighted Poisson estimate (2.2) from [RZ3,RZ4]. That is also central, but it is an external result explicitly cited and flagged as non-Muckenhoupt; without evidence against it I would not make it the primary objection. Since my primary concern differs from the reader's stated weakest_assumption but matches their rationale, agreement is partial. A secondary check worth running is verification of Lemma 2.2 for p=3, μ>2/3 in the cylinder geometry, since (3.18) would collapse if that estimate failed. The present verdict is unchanged from the reader's conditional assessment: the paper likely can be repaired, but as written the stated L^{6/5} assumption does not close the energy estimate.","tokens_in":26080,"tokens_out":11477,"duration_ms":124217,"concrete_test":"To settle whether the gap lands, re-derive the passage from (3.28) to (3.32) under the stated assumption ωf∈L^{6/5}(Ω_T). Specifically, write the inequality used for ∫_Ω ω(ϑ)f·w dx in Lemma 3.2; if it is ||ωf||_{L^{6/5}}||w||_{L^6}, then (3.32) is unjustified. A decisive counterexample to the needed control is g(x)=|x|^{-3/2} on the unit ball Ω⊂R^3: g∈L^{6/5}(Ω) but g∉L^3(Ω), so no constant C can give ||g||^2_{L^3} ≤ C||g||^2_{L^{6/5}}. If the authors instead prove a bound of the form ||ϑ||^2_{L^2}||ωf||^2_{L^3} ≤ C||ωf||^2_{L^{6/5}}||ϑ||^2_{H^1} using the heat equation, that proof must be supplied; as written, the absorption in (3.38) collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Lemma 3.2 jumps from the RHS of (3.28), which contains only |ω(θ)f|^2_{L^{6/5}(Ω)} and boundary terms, to (3.32), whose RHS contains the additional term ||ϑ||^2_{L^2(Ω)}||ω(θ)f||^2_{L^3(Ω)}. No estimate in the preceding lines produces an L^3 norm of ωf; the only available control on the force term in (3.16) is ||ωf||_{L^{6/5}}||w||_{L^6}, which does not involve ϑ. On a bounded 3D domain, membership in L^{6/5} does not imply membership in L^3, so the added term may be infinite. The absorption step in (3.38) is also inconsistent: the paper defines M = ess sup_t ||ωf||^2_{L^{6/5}} and chooses c4 by ¯c2¯c4κ/4 = ¯c1M, but the term that must be absorbed is ||ϑ||^2_{L^2}||ωf||^2_{L^3}; that would require M to be the L^3 norm. Finally, the same missing L^3 regularity enters the Galerkin compactness argument: (4.5) bounds ||∂w_m/∂t||_{L^{4/3}(V_1^*)} using ||f||_{L^{4/3}(L^3)}||ϑ_m||_{L^∞L^2}. Thus the whole existence proof needs f∈L^3, or ωf∈L^3, rather than the stated ωf∈L^{6/5}. This is not a cosmetic typo: it is the mechanism by which the external force is controlled, and it is absent from the theorem's hypotheses.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies viscous incompressible heat-conducting flow in a finite cylinder with large inflow and outflow, modelled by a Navier-Stokes-type system coupled to a heat equation. The authors introduce a Hopf-type correction δ, reduce the velocity to a divergence-free field w with homogeneous boundary data, and then derive an energy estimate in which the nonlinear terms involving δ and ∇δ are controlled by weighted Sobolev estimates from their earlier work. The Main Theorem claims an energy estimate and existence of weak solutions with no restrictions on the magnitudes of the external force, initial data, or boundary flux, assuming only ωf ∈ L^{6/5}(Ω_T), d_i ∈ L^∞(0,T;W^1_3(S_2(a_i))) ∩ L^2(0,T;H^{1/2}(S_2(a_i))), and θ(0) ∈ L^2(Ω). The proof proceeds via a priori estimates for w and ϑ, followed by a Galerkin approximation and compactness arguments.","tokens_in":26476,"tokens_out":5544,"duration_ms":61374,"significance":"If correct, the result would be a substantial extension of earlier large-flux results: it removes magnitude restrictions on the data and weakens conditions on the temperature-dependent coefficient ω(θ), while retaining an explicit energy estimate. The treatment of the nonlinear terms through weighted spaces and the explicit tracking of constants are genuine strengths. However, the contribution is not machine-checked, and the central estimate rests on a gap in the control of the force term: an L^3 norm of ωf appears in the key absorption step and in the Galerkin compactness argument without being available from the stated hypotheses. The significance of the paper can therefore only be assessed after this load-bearing gap is repaired, for instance by adding an L^3-in-space condition on ωf or proving such a bound from the existing assumptions.","major_comments":[{"comment":"The transition from (3.31) to (3.32) is not justified. The right-hand side of (3.31) contains only ||ω(θ)f||^2_{L^{6/5}(Ω)} together with boundary terms involving d, whereas (3.32) contains the additional product ||ϑ||^2_{L^2(Ω)} ||ω(θ)f||^2_{L^3(Ω)}. No estimate in the preceding lines produces an L^3 norm of ω(θ)f, and on a bounded three-dimensional domain membership in L^{6/5} does not imply membership in L^3. If this term is intended to arise from a Hölder estimate on the force integral, the estimate must be written out; otherwise the inequality is not established. This is not a cosmetic issue because this term is exactly what the subsequent absorption step is designed to control.","section":"Section 3, equation (3.32)"},{"comment":"The absorption step after (3.38) is inconsistent. The constant M is defined as ess sup_t ||ω(θ)f||^2_{L^{6/5}(Ω)}, and ¯c4 is chosen so that ¯c2 ¯c4 κ/4 = ¯c1 M. But the term to be absorbed is ||ϑ||^2_{L^2(Ω)} ||ω(θ)f||^2_{L^3(Ω)}. To absorb this by ¯c2 ¯c4 κ/2 ||ϑ||^2_{H^1(Ω)} one would need a bound of the form ||ω(θ)f||^2_{L^3(Ω)} ≤ c M, which is not available under the assumption ωf ∈ L^{6/5}. Even if such a bound existed, the correct constant would involve ess sup ||ω(θ)f||^2_{L^3}, not the L^{6/5} quantity. Hence the displayed energy inequality obtained after (3.38) does not follow from the stated hypotheses.","section":"Section 3, text after equation (3.38)"},{"comment":"The Galerkin compactness argument also requires L^3 control of the force. In the estimate for ∂w_m/∂t in L^{4/3}(kT,(k+1)T;V_1^*) displayed in (4.5), the term ||f||_{L^{4/3}(kT,(k+1)T;L^3(Ω))} ||ϑ_m||_{L^∞(kT,(k+1)T;L^2(Ω))} appears. Under the assumptions of Theorem 4.1, only ωf ∈ L^{6/5} is available, and without additional regularity or boundedness of ω, this term is not controlled. Consequently the uniform bound (4.6), which is needed for the Aubin-Lions compactness argument, is not established. The same missing L^3 regularity therefore affects both the energy estimate and the existence proof.","section":"Section 4, equation (4.5)"},{"comment":"The paper's central smallness mechanism relies on the weighted Poisson estimate (2.2), cited from the authors' previous works [RZ3,RZ4], but the version stated in Lemma 2.2 is too coarse for the use made of it. The application in (3.18)-(3.19) requires a specific range of the weight exponent, namely μ > 2/3, and the weight η^{3μ} is explicitly not a Muckenhoupt weight. Since the control of ||∇φ||_{L^3} in (3.18) is essential for estimating J_2, the proof is not self-contained, and the reader cannot verify from the manuscript that the estimate holds for the required exponents and geometry. If the journal permits citation of the authors' prior monograph and papers, this is acceptable as an external tool, but the precise statement with admissible exponents should be given, and the paper should flag that its main smallness lemma is taken from previous work rather than proved here.","section":"Section 3, equations (3.18)-(3.19) and Lemma 2.2"}],"minor_comments":[{"comment":"Equation (3.49) has unbalanced parentheses: the closing parenthesis inside the exponential is missing, and the displayed right-hand side mixes ||θ(0)||^2_{L^2(Ω)} with ||θ(0)||^2_{L^1(Ω)}. In the proof, the line after (3.50) should read c|d1|^6_{3,S_2(-a)} |θ|^2_{2,Ω}, not |θ(0)|^2_{2,Ω}.","section":"Theorem 3.7, equation (3.49) and its proof"},{"comment":"The proof of Lemma 2.4 establishes bounds on |θ(t)|_{∞,Ω}, but the statement claims pointwise bounds θ_* ≤ θ(t) ≤ θ^*. The passage from the L^∞ norm to pointwise bounds is not explained, and the lower-bound argument only controls 1/|θ(t)|_{∞,Ω}, not the pointwise reciprocal. The statement should be reformulated, or the missing interpolation/trace argument should be provided.","section":"Lemma 2.4"},{"comment":"Several equation references in the proof of Lemma 3.2 are inaccurate. For example, the sentence 'multiply (3.28) by a sufficently large constant ¯c4 and add the result to the inequality (3.21)' appears to refer to the temperature estimate (3.37), not (3.21); and the line 'integrating (3.29) with respect to time' in Lemma 3.4 should refer to (3.40). These cross-reference errors should be corrected.","section":"Proof of Lemma 3.2, equation references"},{"comment":"The manuscript contains several evident typos and malformed displays: the repeated phrase 'NStemp-weak-energy — 16−8−2025' in the page footers, 'Renc/suppress lawowicz' in the author headers, and inconsistent notation for the boundary sets S_2(a_i) versus S_2(ai). These do not affect the mathematics but should be cleaned up before publication.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The L^3-force issue identified in the reader's report is real and is confirmed in equations (3.32), (3.38), and (4.5). I see no grounds for rejection on grounds of novelty or dishonesty: the paper is an honest attempt at a difficult problem, and the central claim could become correct if the hypotheses are strengthened to include, for example, ωf ∈ L^{4/3}(0,T;L^3(Ω)) or an equivalent condition that makes the displayed absorption legitimate. The authors should also be asked to state explicitly which weighted estimates are proven here and which are imported from [RZ2,RZ3,RZ4], since the main smallness mechanism depends on a nonstandard weighted Poisson estimate. I do not think the manuscript is ready in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this before citing: the paper claims an energy estimate for heat-conducting Navier-Stokes with arbitrarily large flux, but the proof of the key estimate has a load-bearing gap. The L^3 term in (3.32) appears from nowhere, and the absorption step after (3.38) uses the wrong norm. The main theorem is not established under the stated assumptions.\n\nWhat's genuinely new and good: Lemma 2.4 gives two-sided L^∞ bounds on temperature with the inflow only in L^2(L∞). That's a real improvement over [ZZ], and it's cleanly proved. The overall strategy—Hopf function plus weighted spaces to absorb the nonlinear terms—is coherent, and the estimates for the boundary terms and δ are worked out in detail. If the force term were fixed, the rest of the mechanism looks plausible.\n\nWhere it breaks: Equation (3.28) has RHS containing |ω(θ)f|^2_{L^{6/5}} and various d-terms. Then (3.32) suddenly contains ||ϑ||^2_{L^2} ||ω(θ)f||^2_{L^3}. No estimate in between produces an L^3 norm of ωf. The only available control on the force is ∫ ωf·w bounded via ||ωf||_{6/5}||w||_6. That gives no ϑ factor and no L^3 norm. The absorption step is also inconsistent: M is defined as ess sup ||ωf||^2_{L^{6/5}}, but the term to be absorbed is ||ϑ||^2 ||ωf||^2_{L^3}; that requires M to be the L^3 norm. The same missing L^3 regularity shows up in (4.5), where the bound on ∂w_m/∂t uses ||f||_{L^{4/3}(L^3)} ||ϑ_m||_{L∞L^2}. So the Galerkin compactness argument needs f∈L^3, or ωf∈L^3, not just ωf∈L^{6/5}. This is the mechanism controlling the external force, and it's absent from the hypotheses.\n\nIs it fixable? Probably yes, by strengthening the assumption to ωf∈L^3 or adding a condition that controls the L^3 factor. But as written, the theorem does not follow. The reliance on Lemma 2.2 from their earlier papers is a separate point; I'd want to check that weighted estimate, but the immediate problem is the force term.\n\nWho's this for: anyone in the large-flux Navier-Stokes program. The temperature bound lemma is worth keeping; the energy estimate needs revision. A serious referee should be assigned because the problem is significant and the gap is repairable, but the paper should not be accepted as is.","headline":"Claims arbitrary-large-flux energy estimate for heat-conducting Navier-Stokes, but a missing L^3 force term in Lemma 3.2 leaves the main theorem unproved as written.","tokens_in":27002,"tokens_out":4441,"would_cite":false,"duration_ms":41683,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","76D03","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the coupled Navier-Stokes/heat system in a finite cylinder with large inflow and outflow admits weak solutions satisfying an energy estimate with no size restrictions on force, initial data, or flux.","keywords":["Navier-Stokes equation","heat-conducting fluid","weak solutions","energy estimate","large flux","weighted Sobolev spaces","Galerkin method","Hopf function"],"falsifier":"Choose a sequence of smooth lid fluxes $d_i$ concentrated in smaller and smaller patches, construct the Hopf carrier $b$ and the Neumann solution $\\phi$ exactly as in the paper, and compute the ratio $\\|\\nabla^2\\phi\\|_{L^{3,\\mu}(\\Omega)}/\\|\\operatorname{div} b\\|_{L^{3,\\mu}(\\Omega)}$ with $\\mu>2/3$. If this ratio is unbounded along the sequence, the weighted estimate (2.2) fails in the regime the proof needs, and the absorption of $J_2$ and $J_4$ in the energy inequality cannot be justified.","tokens_in":25876,"feed_emoji":"🌊","tokens_out":10041,"duration_ms":91326,"temperature":0.7,"pith_summary":"This paper takes on the coupled Navier-Stokes and heat-conduction system for a viscous incompressible fluid in a finite cylinder, with prescribed inflow and outflow through the two flat ends. It proves that weak solutions exist and satisfy a global energy estimate with no size restrictions on the external force, the initial data, or the inflow and outflow rates. The key move is to subtract a flux carrier built from a Hopf cutoff and a weighted Poisson correction, so that the remaining velocity field has homogeneous boundary conditions and is divergence-free; the nonlinear terms are then absorbed using weighted Sobolev norms in which the weight is the distance to the inflow/outflow lids. The temperature is shown, before the coupling is closed, to stay between two positive bounds, which relaxes the hypotheses previously needed on the temperature-dependent force coefficient. If correct, this removes the large-flux restriction for this class of heat-conducting motions and allows the flux to remain large as time grows.","feed_headline":"Heat-conducting flow yields weak solutions under any flux size","feed_subtitle":"An energy estimate for the coupled Navier-Stokes/heat system holds with no size limits on force, data, or boundary flow.","key_machinery":"The central mechanism is the decomposition $v = w + b + \\nabla\\phi$, where $b$ is a Hopf-type flux carrier: a logarithmic cutoff function, supported in thin neighbourhoods of the two lids $S_2(\\pm a)$, chosen so that $b\\cdot\\bar n$ reproduces the prescribed data $d_1,d_2$ while $w$ has zero normal component on the whole boundary. The function $\\phi$ solves the Neumann problem $\\Delta\\phi=-\\operatorname{div} b$ with $\\bar n\\cdot\\nabla\\phi=0$, making $w$ divergence-free as well. Everything turns on the weighted elliptic estimate $\\|\\nabla^2\\phi\\|_{L^{p,\\mu}(\\Omega)}\\le c\\|\\operatorname{div} b\\|_{L^{p,\\mu}(\\Omega)}$, with weight $\\eta(x_3)^{p\\mu}$ where $\\eta$ is the distance to $S_2$; the paper cites this estimate from earlier work and uses it with $p=3$ and $\\mu>2/3$ to control $\\nabla\\phi$ in $L^3(\\Omega)$. That control, together with the smallness parameters $\\varepsilon$ and $\\varrho$ coming from the Hopf cutoff, lets the nonlinear terms $w\\cdot\\nabla\\delta\\cdot w$ and $\\delta\\cdot\\nabla w\\cdot w$ be absorbed into the dissipation terms on the left-hand side of the energy inequality.","core_discovery":"The main theorem claims that under the regularity assumptions $d_i \\in L^\\infty(0,T;W^1_3(S_2(a_i))) \\cap L^2(0,T;H^{1/2}(S_2(a_i)))$, $i=1,2$, $\\theta(0)\\in L^2(\\Omega)$, and $\\omega f \\in L^{6/5}(\\Omega_T)$, the original problem has a weak solution $(v,\\theta)$ in the sense defined through the corrected pair $(w,\\vartheta)$, and the energy estimate (1.3) holds. The estimates are proved on successive time intervals $[kT,(k+1)T]$ with constants controlled by the data through a term $A_1^2$ involving $\\|\\omega(\\theta)f\\|^2_{L^{6/5}}$, $\\|d_i\\|^2_{W^1_3(S_2(a_i))}$ and $\\|d_{i,t}\\|^2_{W^1_{6/5}(S_2(a_i))}$, multiplied by an increasing function of $\\sup_t \\|d_i(t)\\|_{W^1_3(S_2(a_i))}$. The temperature estimate is separate: with $\\theta_0$ bounded between $\\theta_*$ and $\\theta^*$ and $d_1$ in $L^6(0,T;L^3(S_2(-a)))\\cap L^2(0,T;L^\\infty(S_2(-a)))$, the temperature remains in that interval and its $V$-norm satisfies an exponential bound in $\\|d_1\\|^6_{L^6(0,T;L^3(S_2(-a)))}$. Existence is completed by the Galerkin method, using the energy bounds to extract weakly and strongly convergent subsequences on each time interval.","pith_inferences":["A likely extension is to ducts or channels whose ends are not flat: the proof only uses the distance-to-lid weight and the Hopf cutoff, so the same energy structure may survive under mild curvature of the inflow/outflow surfaces.","Because the forcing enters only through $\\omega f \\in L^{6/5}$, the method should tolerate growth of $\\omega$ with $\\theta$ as long as the product stays in that space; the already-proved two-sided bounds on $\\theta$ make this a testable relaxation.","The sharp range of the exponent $\\mu$ in the weighted Neumann estimate determines whether the energy estimate is purely structural or carries a hidden geometric restriction; computing the constant in Lemma 2.2 on a family of thin cylinders would settle this.","One could test the mechanism numerically: build $b$ from a smooth flux $d$, solve the Neumann problem for $\\phi$, and compare $\\|\\nabla^2\\phi\\|_{L^{3,\\mu}}$ with $\\|\\operatorname{div} b\\|_{L^{3,\\mu}}$; any unbounded ratio as the flux is concentrated near the lids would point to a missing assumption."],"forward_implications":["If the main theorem is correct, large boundary flux is no longer an obstruction: the energy estimate contains no smallness condition on $d_1$, $d_2$, $f$, $v_0$, or $\\theta_0$.","The flux is allowed to stay large as $t\\to\\infty$ rather than decaying, because the proof only needs the data to satisfy the stated integrability and regularity bounds on each time interval.","The temperature-dependent force coefficient $\\omega(\\theta)$ can be treated under weaker hypotheses than in the earlier existence theory, since $\\theta$ is shown to be bounded above and below before the term $\\omega(\\theta)f$ is estimated.","The energy estimate is explicit: the velocity and temperature norms are controlled by powers of the flux norm, an exponential factor of the form $\\exp(c\\|d\\|_{W^{1,3,\\infty}})$, and $\\|\\omega(\\theta)f\\|^2_{L^{6/5}(\\Omega_T)}$.","Weak solutions are obtained on every interval $[kT,(k+1)T]$, which gives a global-in-time existence statement under the stated assumptions."],"supporting_citations":[{"why":"Supplies the Hopf function used to construct the flux carrier $b$ matching the prescribed boundary data $d$ on $S_2$.","marker":"[L1]"},{"why":"Book-length development of the large-flux problem; it supplies the construction of the Hopf extension, the Neumann correction, and the weighted estimates inherited by this paper.","marker":"[RZ2]"},{"why":"Proves the weighted Poisson estimate for $L^2$-weighted spaces used in Lemma 2.2 and in the control of $\\nabla\\phi$.","marker":"[RZ3]"},{"why":"Extends the weighted Poisson estimate to $L^p$-weighted spaces, the $p=3$ version of which is the load-bearing step (3.18).","marker":"[RZ4]"},{"why":"Earlier existence result for the same system with stronger restrictions on $\\omega(\\theta)$; this paper relaxes those conditions and uses its Galerkin strategy.","marker":"[ZZ]"},{"why":"Supplies the Aubin-Lions compactness lemma used to pass from Galerkin approximations to weak solutions.","marker":"[L]"}],"fun_headline_variants":["Heat-conducting flow admits weak solutions under any flux","No flux-size limits for heat-conducting Navier-Stokes weak solutions","Large flux no bar to weak solutions in heat-conducting fluid","Weak solutions proven for heat-conducting flow with large flux","Heat-conducting motions: weak solutions without flux restrictions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on a weighted second-derivative bound for the Neumann correction that is cited from the authors' earlier papers and not reproved here; if that bound fails at the exponent $p=3$ for the cylindrical geometry, the nonlinear terms cannot be absorbed and the energy estimate collapses.","fun_headline_variants_meta":{"raw":{"variants":["Heat-conducting flow admits weak solutions under any flux","No flux-size limits for heat-conducting Navier-Stokes weak solutions","Large flux no bar to weak solutions in heat-conducting fluid","Weak solutions proven for heat-conducting flow with large flux","Heat-conducting motions: weak solutions without flux restrictions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1506,"prompt_tokens":991,"completion_tokens":515,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":427}},"tokens_in":607,"tokens_out":515,"duration_ms":5105,"temperature":1.0,"reasoning_tokens":427,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:09:40.486409+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a sequence of smooth lid fluxes $d_i$ concentrated in smaller and smaller patches, construct the Hopf carrier $b$ and the Neumann solution $\\phi$ exactly as in the paper, and compute the ratio $\\|\\nabla^2\\phi\\|_{L^{3,\\mu}(\\Omega)}/\\|\\operatorname{div} b\\|_{L^{3,\\mu}(\\Omega)}$ with $\\mu>2/3$. If this ratio is unbounded along the sequence, the weighted estimate (2.2) fails in the regime the proof needs, and the absorption of $J_2$ and $J_4$ in the energy inequality cannot be justified.","supporting_citations":[],"review_version":1}