{"id":"b973f967-5a6a-4f04-91a8-52fcbee31234","arxiv_id":"2608.11553","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"The forced 3D Navier-Stokes equations admit a pure-swirl solution with force in L^1_t L^2_x whose L infinity norm tends to infinity at the final time, and the exact mixed-norm ranges are determined.","lead":"These authors construct an explicit swirling flow in a cylinder whose velocity grows without bound at a chosen final time even though the driving force is finite enough to lie in the energy class L^1_t L^2_x. A generalist should care because the result pins down the sharp integrability range in which forced Navier-Stokes can still lose boundedness.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the pure-swirl reduction is a disclosed scope limitation, and the construction, rates, and mixed-norm classifications are internally consistent.","rationale":"The reader identified the pure-swirl/free-slip ansatz as the weakest assumption; I agree that this is the only fragility, but it is a scope limitation the authors disclose, not an error. I verified the algebraic core: the profile lemma, the scaling identity, the force definition, the L^1_tL^2_x integrability exponent, and the boundary and axis smoothness. The mixed-norm equivalences follow by integration of the two-sided rates. No unproven step beyond standard weak-strong uniqueness appears, and that step is cited. Accordingly the verdict should stand unchanged.","tokens_in":9800,"tokens_out":50430,"duration_ms":483844,"concrete_test":"Recompute the forcing identity (31) via the product rule L(r^αφ)=r^αLφ+α^2r^{α-2}φ+2αr^{α-1}∂rφ and then integrate the m=2 case of (32): if ∥f(t)∥_{L^2} does not behave as (T-t)^{α/2-1}, the claimed f∈L^1_tL^2_x and the main theorem would collapse. This single check separates an algebra slip from a genuine construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After line-by-line checking, the central construction withstands scrutiny. The profile identity (22) follows from g=ρφ0; the rescaled identity (26) has the correct sign since R_t=-1/R; and the forcing (31) is exactly ∂tW-LW because L(r^αφ)=r^αLφ+α^2r^{α-2}φ+2αr^{α-1}∂rφ. Lemma 4.1 exponents are correct: with B_m=(3-α)m/2-1, (32) gives ∥f(t)∥_{L^2}≍(T-t)^{α/2-1}, which is L^1_t-integrable precisely for α>0. The lower bounds use the exact outer profile W=β(r-r^{α-1}), and the inner cutoff A∈C_c^∞((a,b)) makes all fields C^∞ on D×[0,T). Boundary conditions and the strong L^2 terminal trace are handled correctly. The only substantive caveat is that pure swirl makes convection a pure pressure gradient; the paper states this explicitly in Remark 1.1 and does not overclaim nonlinearity. I therefore do not identify a load-bearing correctness concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for each T>0 and each finite p,q with 1/p+1/q>1, an explicit pure-swirl solution of the forced three-dimensional Navier-Stokes equations in a cylinder with mixed boundary conditions. The force lies in L^q(0,T;L^p(D)) and automatically also in L^1(0,T;L^2(D)); the solution is classical on [0,T), has finite energy, satisfies the energy equality, extends strongly in L^2 to the unique Leray-Hopf solution at T, and has L-infinity norm diverging as t approaches T. The construction refines Zhang's profile with an annular cutoff in the self-similar variable, and the authors derive two-sided asymptotic rates for the force, the velocity supremum norm, and the enstrophy, together with exact mixed-norm thresholds for force and velocity.","tokens_in":9993,"tokens_out":24994,"duration_ms":235059,"significance":"If the proof is correct, the paper provides a sharp, explicit example showing that L^1_t L^2_x forcing is compatible with terminal loss of boundedness within the Leray-Hopf class, with uniqueness and energy equality holding. The argument is explicit and checkable: the profile identity (22), the rescaled parabolic identity (26), the commutator computation leading to (31), and the two-sided estimates in Lemmas 4.1-4.3 are all consistent. A notable strength is that the construction is not an abstract existence result but a fully explicit family with exact rates. The main limitation, clearly acknowledged in Remark 1.1, is that the pure-swirl ansatz makes the convective term a pure pressure gradient, so the mechanism is linear and the example also solves the forced Stokes system; this does not undermine the stated claims but tempers the significance for genuinely nonlinear Navier-Stokes dynamics.","major_comments":[],"minor_comments":[{"comment":"The references to 'theorem 4.1', 'theorems 4.1 and 4.2', 'theorem 4.3', and 'theorem 4.4' should read 'Lemma 4.1', 'Lemmas 4.1 and 4.2', 'Lemma 4.3', and 'Proposition 4.4', respectively.","section":"Proof of Theorem 1.2"},{"comment":"References [11] (Giga) and [22] (Solonnikov) do not appear to be cited in the text; the authors should either cite them where relevant or remove them from the bibliography.","section":"References"},{"comment":"The sentence 'The large-y behavior of the integrand is y^{1-m(1-alpha)}' could be clarified by stating that the integral over [0,1/sqrt(s)] converges, diverges logarithmically, or diverges as a power according as m(1-alpha)<2, =2, or >2; the current wording is correct but telegraphic.","section":"Lemma 4.3"},{"comment":"There are several typographical and formatting issues in the rendering of mixed-norm notation (e.g., 'L1tL2 x'); these should be fixed in the final version.","section":"Title and Abstract"},{"comment":"The paragraph after Lemma 4.3 would be better placed as a formal remark, since it contains substantive qualitative information about concentration on the symmetry axis.","section":"Section 4"}],"recommendation":"minor_revision","confidential_remarks":"This is a solid, fully explicit construction. The main caveat for the editor is that, because the convective term is exactly canceled by the pressure in the pure-swirl ansatz, the result is essentially a statement about the forced linear Stokes/transport equation; the authors are transparent about this. The paper is closely tied to the authors' prior work [2], and the incremental novelty beyond [2] and Zhang [23] should be weighed by the editors, though the corrected L^1_tL^2_x endpoint and the two-sided rates are clearly added value."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one if you work on forced Navier–Stokes singularities. The paper sharpens Zhang's construction: for every T>0 and 1/p+1/q>1, it builds a smooth pure-swirl forced solution in a cylinder whose L∞ norm blows up at T, with the force in L^q_t L^p_x and always in L^1_t L^2_x. The genuinely new pieces are the corrected endpoint condition (the missing p-th root from their own weighted paper), the annular cutoff that keeps noninteger powers of r smooth at the axis, and the two-sided asymptotics for force, velocity, and enstrophy.\n\nThe proof is explicit and line-by-line checkable; the profile identity (22), the force identity (31), and the exponent bookkeeping in Lemmas 4.1–4.3 are consistent. The terminal trace, energy equality, and uniqueness argument rely on standard Serrin-type weak–strong uniqueness, cited rather than re-derived; that's acceptable for a note of this kind.\n\nThe soft spot is the one the authors name. Pure swirl makes the convective term purely radial, so it is absorbed into the pressure and the evolution is a scalar linear parabolic equation. The example is therefore a sharp counterexample for the forced Stokes system as much as for Navier–Stokes; it does not touch the unforced regularity problem. That limits significance but is not a flaw in the statement. Also, the force is defined from the solution, which is legitimate for an existence theorem, but the result only describes what this special profile can see, not a general forced-data statement.\n\nMinor quibble: the velocity mixed-norm classification is exact for this construction, so the 'exact ranges' phrasing is conditional on the pure-swirl ansatz. The paper says this, but a reader skimming the abstract might miss it.\n\nCitation pattern is fine: Zhang's profile is credited, and their earlier work is explicitly corrected rather than hidden. No code or data applies. This paper is for people working on forced blow-up, counterexamples to regularity via external forces, and Serrin-class uniqueness. It deserves a serious referee; I would send it to review and expect acceptance after minor polish, with the endpoint discussion double-checked.","headline":"A checkable, honest sharpening of Zhang's forced blow-up construction; the pure-swirl caveat is real but disclosed.","tokens_in":10556,"tokens_out":3090,"would_cite":true,"duration_ms":33563,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","76D03","35A01"],"pacs":[],"model":"deepseek-v4-flash","headline":"A force in $L^1(0,T;L^2(D))$ can make a smooth, finite-energy Navier–Stokes solution lose boundedness at a finite time.","keywords":["Navier–Stokes equations","external force","loss of boundedness","pure swirl","mixed norms","Leray–Hopf solution"],"falsifier":"Take the explicit formulas (30)–(31), fix an admissible $\\alpha$, and check the two-sided bound (32) numerically or symbolically: the $L^q(0,T;L^p(D))$ norm of $f$ must be finite exactly when $\\alpha>3-2/p-2/q$, with logarithmic divergence at equality. The decisive identity to verify is $F(r,t)=-\\beta\\alpha(2-\\alpha)r^{\\alpha-3}$ on the annular shell $bR(t)\\le r\\le r_0$; a single counterexample to this identity or to the stated rate would falsify Lemma 4.1 and with it Theorem 1.2.","tokens_in":9566,"feed_emoji":"🌀","tokens_out":13826,"duration_ms":152143,"temperature":0.7,"pith_summary":"The paper constructs a concrete example showing that an external force of energy-class integrability -- $f\\in L^1(0,T;L^2(D))$ -- can drive a smooth, finite-energy solution of the forced three-dimensional Navier–Stokes equations in a cylinder to lose boundedness: $\\|v(t)\\|_{L^\\infty(D)}\\to\\infty$ as $t\\uparrow T$. The construction works for every pair of exponents $1\\le p,q<\\infty$ with $1/p+1/q>1$, and it determines the exact mixed-norm ranges of the force and the velocity for this family. Since the example is classical before $T$, extends to a unique Leray–Hopf solution, and satisfies the energy equality up to $T$, the paper shows that these standard guarantees do not exclude terminal loss of boundedness. The same explicit fields solve the forced Stokes system, because the convection term is exactly absorbed by the pressure.","feed_headline":"A smooth, finite-energy force can make a fluid flow lose boundedness","feed_subtitle":"An explicit swirl example shows the velocity's supremum norm diverges exactly at the final time.","key_machinery":"The load-bearing object is a pure-swirl velocity field $v=W(r,t)e_\\theta$ in the circular cylinder, so the fluid moves only around the axis and $W$ depends only on radius and time. In that ansatz $(v\\cdot\\nabla)v=-W^2r^{-1}e_r$ is a gradient, so it is absorbed into the pressure and the azimuthal equation reduces to the scalar linear heat-type equation (17): $\\partial_tW-(\\partial_r^2+r^{-1}\\partial_r-r^{-2})W=F$. The solution is built from a self-similar profile $\\phi_0$, Zhang's profile refined by an annular cancellation: $\\phi_0\\equiv0$ for $r\\le a$ and $\\phi_0=-\\beta/r$ for $r\\ge b$, thanks to a compactly supported weight $A\\in C_c^\\infty((a,b))$. Setting $R(t)=\\sqrt{2(T-t)}$, the velocity is $W(r,t)=r^\\alpha R(t)^{-1}\\phi_0(r/R(t))+\\beta r$, and the force is chosen as $F=-r^\\alpha h-\\alpha^2r^{\\alpha-2}\\phi-2\\alpha r^{\\alpha-1}\\partial_r\\phi$. The annular cancellation keeps the non-integer factor $r^\\alpha$ smooth at the axis; the exact outer identity $F=-\\beta\\alpha(2-\\alpha)r^{\\alpha-3}$ on $bR(t)\\le r\\le r_0$ and the pointwise bounds (27) yield all the two-sided rates.","core_discovery":"The paper's central claim is that terminal loss of boundedness is compatible with every standard regularity guarantee short of the sup norm. Concretely, for any $T>0$ and any $1\\le p,q<\\infty$ with $1/p+1/q>1$, Theorem 1.2 provides a force $f\\in L^q(0,T;L^p(D))\\cap L^1(0,T;L^2(D))$ and a classical solution $(v,P)$ of (1)–(2), smooth on $D\\times[0,T)$, such that $\\|v(t)\\|_{L^\\infty(D)}\\to\\infty$ as $t\\uparrow T$ while $v\\in L^\\infty(0,T;L^2(D))\\cap L^2(0,T;H^1(D))$. The solution extends strongly in $L^2$ to the unique Leray–Hopf solution on $[0,T]$ and satisfies the energy equality. The construction gives two-sided rates, e.g. $\\|v(t)\\|_{L^\\infty(D)}\\asymp(T-t)^{-(1-\\alpha)/2}$ and $\\|f(t)\\|_{L^p(D)}\\asymp(T-t)^{-(3-\\alpha)/2+1/p}$ for a parameter $\\alpha$ chosen in (8). It also classifies the mixed norms exactly: within this family $f\\in L^q(0,T;L^p(D))$ holds precisely for $\\alpha>3-2/p-2/q$, and $v\\in L^\\sigma(0,T;L^m(D))$ precisely for $\\alpha>1-2/m-2/\\sigma$.","pith_inferences":["The thresholds are exact for the constructed pure-swirl family; the paper does not prove that every force with $1/p+1/q>1$ must produce loss of boundedness, nor that exponents with $1/p+1/q\\le1$ forbid it.","Because the mechanism is a linear scalar equation dressed by pressure, the annular-profile construction is likely transplantable to other linear or semilinear parabolic problems, and a testable extension would be to run the same construction in a ball instead of a finite cylinder.","The loss of boundedness concentrates on the symmetry axis at spatial scale $\\sqrt{T-t}$; an instructive perturbation would add a small radial or axial velocity component and check whether this cancellation, and the mixed-norm thresholds, survive."],"forward_implications":["For every finite $p\\ge1$ the construction allows $q=1$, so the force can always be taken in $L^1(0,T;L^2(D))$; energy-class integrability of the forcing is compatible with unboundedness of the velocity.","The mixed-norm classification is exact for this family: $f\\in L^\\sigma(0,T;L^m(D))$ holds precisely when $\\alpha>3-2/m-2/\\sigma$, with logarithmic divergence at equality; in particular the advertised $L^1_t L^2_x$ case is included.","One may choose the parameter $\\alpha$ so that the velocity lies in any prescribed $L^\\sigma(0,T;L^m(D))$ while still having $\\|v(t)\\|_{L^\\infty(D)}\\to\\infty$.","The extended solution is unique in the Leray–Hopf class and satisfies the energy equality on $[0,T]$, so a sup-norm loss of boundedness does not force nonuniqueness or an energy defect.","Because the nonlinear term is absorbed into the pressure, the same example is a solution of the forced Stokes system, showing the effect is essentially linear."],"supporting_citations":[{"why":"Supplies the self-similar scalar profile formula (20) that the present construction refines with an annular cancellation.","marker":"[23]"},{"why":"The authors' previous note; it supplies the pure-swirl reduction observation and the weighted $r^\\alpha\\phi$ ansatz, and its endpoint discussion is superseded here.","marker":"[2]"},{"why":"Serrin's weak–strong uniqueness theory underpins the uniqueness argument in Proposition 4.4.","marker":"[20]"},{"why":"Provides the extension of Serrin-type estimates quoted for the weak–strong difference bound (52).","marker":"[7]"},{"why":"Kozono–Sohr uniqueness result is one of the references used to justify the weak–strong difference estimate.","marker":"[14]"},{"why":"Rules out nontrivial unforced backward self-similar solutions, making the singular forcing essential to the mechanism.","marker":"[18]"}],"fun_headline_variants":["Smooth finite-energy force drives swirl to blow-up","Finite-energy smooth forcing yields terminal blow-up","Explicit swirl blow-up under mild forcing","Benign smooth force, yet swirl blows up"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the fluid moves only around the cylinder's axis (purely azimuthal flow, $v_r=v_3=0$) with free-slip horizontal boundaries; this makes $(v\\cdot\\nabla)v$ a pure pressure gradient and reduces the evolution to the scalar linear equation (17), and the construction does not go through without this kinematic and boundary ansatz.","fun_headline_variants_meta":{"raw":{"variants":["Smooth finite-energy force drives swirl to blow-up","Finite-energy smooth forcing yields terminal blow-up","Explicit swirl blow-up under mild forcing","Benign smooth force, yet swirl blows up"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000816,"raw_usage":{"total_tokens":3672,"prompt_tokens":1140,"completion_tokens":2532,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":756,"completion_tokens_details":{"reasoning_tokens":2473}},"tokens_in":756,"tokens_out":2532,"duration_ms":25038,"temperature":1.0,"reasoning_tokens":2473,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:36:10.588686+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the explicit formulas (30)–(31), fix an admissible $\\alpha$, and check the two-sided bound (32) numerically or symbolically: the $L^q(0,T;L^p(D))$ norm of $f$ must be finite exactly when $\\alpha>3-2/p-2/q$, with logarithmic divergence at equality. The decisive identity to verify is $F(r,t)=-\\beta\\alpha(2-\\alpha)r^{\\alpha-3}$ on the annular shell $bR(t)\\le r\\le r_0$; a single counterexample to this identity or to the stated rate would falsify Lemma 4.1 and with it Theorem 1.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the self-similar scalar profile formula (20) that the present construction refines with an annular cancellation."},{"cited_title":"Beir˜ ao da Veiga and J","cited_arxiv_id":null,"evidence_quote":"The authors' previous note; it supplies the pure-swirl reduction observation and the weighted $r^\\alpha\\phi$ ansatz, and its endpoint discussion is superseded here."},{"cited_title":"Serrin,The initial value problem for the Navier–Stokes equations, in: Nonlinear Problems (Proc","cited_arxiv_id":null,"evidence_quote":"Serrin's weak–strong uniqueness theory underpins the uniqueness argument in Proposition 4.4."},{"cited_title":"Farwig, H","cited_arxiv_id":null,"evidence_quote":"Provides the extension of Serrin-type estimates quoted for the weak–strong difference bound (52)."},{"cited_title":"Kozono and H","cited_arxiv_id":null,"evidence_quote":"Kozono–Sohr uniqueness result is one of the references used to justify the weak–strong difference estimate."},{"cited_title":"Neˇ cas, M","cited_arxiv_id":null,"evidence_quote":"Rules out nontrivial unforced backward self-similar solutions, making the singular forcing essential to the mechanism."}],"review_version":1}