{"id":"ff10f104-7493-4659-99cb-66161f7c15ad","arxiv_id":"2607.24635","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"For 0<α<1 and 0<s<1-2α+3/p, 3D hypodissipative Navier-Stokes exhibits same-space norm inflation in W^{s,p} and B^s_{p,q}.","lead":"Norm inflation is proved for fractional (hypodissipative) Navier-Stokes in three dimensions: arbitrarily small smooth data grow arbitrarily large in the same Sobolev or Besov space, in arbitrarily short time. The result covers every dissipation power 0<α<1 and the full positive-regularity supercritical range.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the fractional dissipative residual estimate in Lemma 3.9 is the key new step, and it is adequately supported by the interpolation and scale identities in the text.","rationale":"The reader identified the fractional dissipative residual and the strict supercritical gap as the weakest assumption, and my independent review converges on the same point. I checked the derivation of Lemma 3.9 carefully: the reduction of ∇^k E_diss to Λ^{k+2α}ū via bounded Riesz transforms is correct; the interpolation inequality is standard; and the use of Lemma 3.7 supplies the correct scale with constants that may depend on ζ but not on μ. Since all parameters are chosen so that ζ is fixed before μ, the asymptotic statement (3.39) is valid. The stability argument likewise closes: the energy estimates discard only nonnegative dissipative terms, the commutator estimates are standard, and the exponent bookkeeping in Proposition 5.5 and Lemma 5.7 is consistent. I also verified the scale identities underlying the norm growth: K_* = ζ^{-N}μ and A K_*^s V_p = ζ^{2-Ns}, and the error estimates are o(1) relative to this growth. The paper explicitly notes in Remark 1.2 that the construction is limited at δ=0, which is outside the theorem. No internal inconsistency, hidden circularity, or unsupported hypothesis was found. Therefore I see no reason to change the reader's ACCEPT verdict.","tokens_in":17259,"tokens_out":59266,"duration_ms":530401,"concrete_test":"Independently re-derive (3.34) and (3.41) while keeping all ζ-dependence explicit: verify that μ^{2α}/(Aν)=ζ^{-2}μ^{-δ+b(1+1/p)} and that the interpolation constant in (3.41) is O(ζ^{-N(k+2α)}), so that for fixed ζ the residual ratio still tends to 0 as μ→∞.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the argument in good faith. The central claim would fail if the full-space fractional residual were not perturbative, or if the strict supercritical gap were insufficient. The proof, however, is internally consistent. Lemma 3.9 bounds ||∇^k E_diss||_{L^j} by interpolation between L^j and W^{m,j}, which is valid for functions with the thin-torus profile and phase frequency bounded by C_ζ μ on [0,t_*]. The implicit constants may depend on ζ through ζ^{-N}, but ζ is fixed before μ→∞, so this dependence does not affect the asymptotic smallness. The scale identities in Lemma 3.6 give μ^{2α}/(Aν)=ζ^{-2}μ^{-δ+b(1+1/p)}→0 and t_*μ^{2α}→0, exactly as asserted. Remark 1.2 correctly notes that the present balance is lost at δ=0, but that endpoint is outside the theorem's hypotheses. I found no circular dependency, no missing hypothesis in the stability argument, and no contradiction with the local theory. In particular, the bootstrap in Section 5 uses only the projected equation with nonnegative dissipation, and the induction in Proposition 5.5 closes with the stated scale exponents. The Besov and Sobolev norm-growth lower bounds both follow from the non-cancellation lemma without requiring additional regularity. The conclusion is that the central claim is supported by the presented argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves same-space norm inflation for the three-dimensional hypodissipative Navier--Stokes equations with dissipation (-Delta)^alpha, 0<alpha<1. Theorem 1.1 states that for 2<=p<infinity and 0<s<1-2alpha+3/p, in every space X=W^{s,p} or X=B^s_{p,q} with fixed q, there exist divergence-free C_c^infinity initial data of arbitrarily small X-norm whose unique local smooth solution grows to arbitrarily large X-norm in arbitrarily short time. The proof adapts Luo's vortex-ring construction: Section 3 builds a thin-torus approximate solution whose transported swirl develops a phase frequency K_*=zeta^{-N}mu, Section 4 proves the resulting Besov and Sobolev norm growth, and Section 5 establishes stability of the approximation through a Lipschitz bootstrap and an anisotropic H^k induction. The principal new ingredient is Lemma 3.9, which controls the full-space fractional dissipative residual by interpolation and shows that it is perturbative thanks to the strict supercritical gap delta>0. Section 6 fixes the parameters zeta and mu and proves Theorem 1.1.","tokens_in":17521,"tokens_out":5591,"duration_ms":50757,"significance":"If correct, the result substantially extends positive-regularity norm inflation from the classical Navier--Stokes equations to the entire hypodissipative range 0<alpha<1, reaching the full strictly supercritical band below the scaling line s_c(p,alpha)=1-2alpha+3/p. The proof is unusually explicit: the scale identities (3.33)-(3.34), the non-cancellation Lemma 4.2, the projected commutator Lemma 5.2, and the induction in Proposition 5.5 are all written out with precise exponents. A notable strength is the honest limitation statement in Remark 1.2: at the endpoint delta=0 the construction loses both the inflation-time separation and the dissipative smallness, and the paper does not claim endpoint ill-posedness. The theorem implies that the smooth solution map is not locally bounded at the origin in any of the stated supercritical spaces, a strong ill-posedness phenomenon with smooth compactly supported data.","major_comments":[],"minor_comments":[{"comment":"The interpolation argument is sound, but the sentence beginning 'Choose any integer m>k+2alpha' should explicitly state that the interpolation constants are independent of mu and nu once the fixed threshold depending on zeta is passed; this is implicit in Lemma 3.7 but worth making explicit.","section":"Section 3.5, Lemma 3.9"},{"comment":"In the display 'dx=rdrdthetadz=(R+xi/mu) mu^{-2} dxi dtheta dy', the factor 2pi from the theta-integration is omitted; the final estimate is correct because the constant absorbs it, but the presentation would be clearer if this factor were written explicitly.","section":"Section 4.2, Step 4"},{"comment":"The notation ||grad^m u||_{L^infty} is used for the sum over all derivatives of order m; this convention should be stated when it is first introduced, since the displayed estimate is otherwise ambiguous.","section":"Section 5.2, Corollary 5.4"},{"comment":"The exponent arithmetic leading to (3.33) and (3.34) is correct, but adding one line of algebra showing the cancellation in the exponents would help the reader verify the strict negativity for b=delta/100.","section":"Section 3.5, Lemma 3.6"}],"recommendation":"accept","confidential_remarks":"I have no significant confidential concerns. The construction follows Luo's vortex-ring mechanism from the cited literature, and the adaptation to fractional dissipation with the global residual estimate is a genuinely new contribution. The paper is well within the scope of math.AP and the proof appears internally consistent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result, not a routine extension. The paper proves same-space norm inflation for 3D hypodissipative Navier–Stokes with dissipation (−Δ)^α, 0<α<1, in W^{s,p} and B^s_{p,q} for the full strictly supercritical range. The mechanism is Luo's vortex-ring mixing; the genuinely new piece is the treatment of the nonlocal dissipative error on the whole space. The estimate in Lemma 3.9, controlling (−Δ)^α ū by interpolation between L^j and W^{m,j}, is the load-bearing step, and it is written out with the right scale identities. The strict gap δ>0 does the double duty of separating the inflation time from the dissipative time scale and making μ^{2α}/(Aν) negligible. I checked the scale balances in Sections 3 and 5; they are consistent.\n\nThe paper is also honest about its limits. Remark 1.2 states plainly that the construction loses both gains at the critical line δ=0, and that this is a limitation of the construction, not a conclusion about the endpoint. That is the right framing.\n\nSoft spots, in proportion. The Besov and Sobolev lower bounds follow Luo's two preprints for the original vortex-ring picture, though the present paper writes out the non-cancellation and finite-difference lemmas essentially self-contained. It is a mild concern that the foundation includes an accepted and a submitted preprint, but not a red flag. The stability section is dense, with several moving parameters (ζ, μ, ν, K_b, η, ℓ), so it will take a patient referee. I do not see a gap: the H^k induction in Proposition 5.5 closes with the stated exponents, and the Lipschitz closure uses the anisotropy to get μ^η ϑ^{γ_*−1/2}→0.\n\nOne small point worth checking in refereeing is that the interpolation used in Lemma 3.9 is standard but is cited only through Stein for Riesz transforms, not for the fractional interpolation inequality itself. A referee may want a precise reference or a one-line proof. That is minor.\n\nWho this is for: anyone working on ill-posedness of Navier–Stokes and fractional dissipation. It closes a natural gap and gives a template for how the vortex-ring mechanism interacts with nonlocal operators. My recommendation: send it to a serious referee. This is not a desk-reject; it deserves careful checking, with likely acceptance after minor-to-moderate revision.","headline":"Solid adaptation of Luo's vortex-ring mechanism that extends positive-regularity norm inflation to the full hypodissipative range; the strict supercritical gap is used exactly where it should be.","tokens_in":18104,"tokens_out":5456,"would_cite":true,"duration_ms":48850,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35Q35","76D05","35B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves norm inflation for the 3D hypodissipative Navier–Stokes equations with fractional dissipation: arbitrarily small smooth initial data can become arbitrarily large in the same positive-regularity Sobolev or Besov norm in…","keywords":["hypodissipative Navier–Stokes equations","norm inflation","strong ill-posedness","vortex-ring mixing","fractional dissipation","Sobolev spaces","Besov spaces","supercritical regularity"],"falsifier":"The most direct check is quantitative: for the vortex-ring ansatz of Section 3, evaluate the full residual bound (3.28) by computing the ratio $\\|\\nabla^k E(t)\\|_{L^j}/(S_{k,j}A\\nu)$ as $\\mu\\to\\infty$ with the structural parameter $\\zeta$ fixed. Lemma 3.9 predicts this ratio tends to zero precisely when $\\delta>0$; if an explicit evaluation of the torus-profile integrals for any in-range $\\alpha,p,s$ showed the ratio bounded away from zero, the stability Proposition 5.1 would fail and the approximate solution would not stay close to the exact solution on $[0,t_*]$. At the endpoint $\\delta=0$ the same calculation gives $\\mu^{2\\alpha}/(A\\nu)=\\zeta^{-2}$, matching the obstruction recorded in Remark 1.2.","tokens_in":17032,"feed_emoji":"🌀","tokens_out":8863,"duration_ms":74565,"temperature":0.7,"pith_summary":"The paper proves that the three-dimensional hypodissipative Navier–Stokes equations with dissipation $(-\\Delta)^\\alpha$, $0<\\alpha<1$, admit norm inflation in every positive-regularity supercritical space: for any $2\\le p<\\infty$ and $0<s<1-2\\alpha+3/p$, there exist divergence-free, compactly supported, smooth initial data arbitrarily small in $W^{s,p}$ (or $B^s_{p,q}$), whose unique smooth solution becomes arbitrarily large in the same space in an arbitrarily short time. This is a same-space failure of local boundedness for the smooth solution map at the origin. The construction adapts an anisotropic vortex-ring mixing mechanism to the nonlocal dissipation, and the strict supercritical gap $\\delta=1-2\\alpha+3/p-s>0$ is what makes the curvature error and the fractional dissipative error both perturbative. If correct, the result closes the positive-regularity norm-inflation question over the full hypodissipative range, matching what was known for the classical $\\alpha=1$ case.","feed_headline":"Vortex rings make tiny 3D fluid data blow up arbitrarily fast","feed_subtitle":"Proof covers every positive-regularity Sobolev or Besov space below the scaling line.","key_machinery":"The load-bearing object is an anisotropic vortex-ring ansatz: a thin torus of transverse thickness $\\mu^{-1}$ and radius $\\nu^{-1}$ with $\\mu\\gg\\nu$, carrying a meridional two-dimensional steady Euler flow and a swirl whose phase is $\\Phi(t,\\rho,\\varphi)=\\varphi-tA/\\rho$. At the inflation time $t_*=\\zeta^{-N}L^{-1}$, with $L=A\\mu$, the phase develops frequency $K_*=t_*A\\mu^2=\\zeta^{-N}\\mu$, a prescribed large multiple of the original transverse frequency; this converts slow spatial variation into oscillatory high-frequency growth. The new fractional ingredient is the estimate of the nonlocal dissipative residual $E_{\\mathrm{diss}}=(-\\Delta)^\\alpha\\bar u$ by Fourier multipliers: choosing any integer $m>k+2\\alpha$ and interpolating between order $0$ and order $m$ in $L^j$ gives $\\|\\nabla^k E_{\\mathrm{diss}}\\|_{L^j}\\lesssim S_{k,j}\\mu^{2\\alpha}$, and the strict gap $\\delta>0$ makes $\\mu^{2\\alpha}/(A\\nu)=\\zeta^{-2}\\mu^{-\\delta+b(1+1/p)}\\to0$. Thus both the geometric error, which gains the aspect ratio $\\nu/\\mu$, and the nonlocal error are perturbative relative to the derivative scale $S_{k,j}A\\nu$.","core_discovery":"On its own terms, the central claim is Theorem 1.1: for every $0<\\alpha<1$, $2\\le p<\\infty$, and $0<s<1-2\\alpha+3/p$, with $X=W^{s,p}$ or $X=B^s_{p,q}$ and $1\\le q\\le\\infty$, equation (1.1) has the norm-inflation property. Precisely, for every $\\varepsilon>0$ there exists a divergence-free $u_0\\in C_c^\\infty(\\mathbb{R}^3)$ with $\\|u_0\\|_X\\le\\varepsilon$ and a time $0<t_\\varepsilon\\le\\varepsilon$ such that the unique smooth solution exists on $[0,t_\\varepsilon]$ and satisfies $\\|u(t_\\varepsilon)\\|_X\\ge\\varepsilon^{-1}$. The same norm measures the datum and the inflated solution, so the statement is not a cross-space transfer. The paper argues that the vortex-ring mixing mechanism, previously used for the classical Navier–Stokes and Euler equations, survives fractional dissipation across the whole range $0<\\alpha<1$, and that the strict scaling-supercritical gap is precisely the condition that makes the full-space fractional residual negligible relative to the geometric error. Along the way it also proves the Besov version for every fixed summability index $q$ and the Sobolev version by homogeneous interpolation.","pith_inferences":["A natural next question the paper leaves implicit is whether the same vortex-ring mechanism can produce norm inflation at the critical line $s=s_c$ where $\\delta=0$; the paper's own estimates identify the nonlocal residual as the obstruction, so one test is whether smaller aspect ratios or anisotropic fractional dissipation can restore smallness.","The mechanism is geometric rather than Fourier-cascade based, so it may transfer to other equations with fractional dissipation and a transport structure, such as surface quasi-geostrophic or magnetohydrodynamic models, whenever an analogous strict supercritical gap controls the nonlocal term; this is an extrapolation, not a claim of the paper.","If norm inflation holds at every point of the supercritical strip, then the hypodissipative system behaves like the classical Navier–Stokes equations from the point of view of strong ill-posedness, strengthening the case that positive-regularity well-posedness cannot extend below the scaling line.","The stability argument suggests the blow-up is driven by transport and oscillation rather than by energy concentration, so the inflated profile is approximately a high-frequency swirl; this might be observable in numerical simulations as rapid toroidal winding before any singularity forms."],"forward_implications":["The smooth solution map for (1.1) is not locally bounded at the origin in any $W^{s,p}$ or $B^s_{p,q}$ with $0<s<1-2\\alpha+3/p$; in particular it cannot be continuous there.","Norm inflation occurs through genuinely smooth, compactly supported, divergence-free data, so the pathology is not an artifact of weak solution classes or low-regularity spaces.","The full hypodissipative range $0<\\alpha<1$ is covered with one mechanism, matching the classical case $\\alpha=1$; the same torus geometry formally includes $\\alpha=1$.","Because the statement is same-space and the Besov index $q$ is fixed, the result applies to a natural scale of function spaces rather than to one specially chosen norm.","At the scaling line $s=s_c(p,\\alpha)$ the construction loses its perturbative control at $\\delta=0$, so the proof leaves endpoint well-posedness or ill-posedness open."],"supporting_citations":[{"why":"Supplies the anisotropic vortex-ring mixing mechanism and the positive-regularity norm-inflation construction for the classical equations that this paper adapts to fractional dissipation.","marker":"[11]"},{"why":"Develops the Besov picture and the sharp same-space inflation statement that the present theorem extends to $0<\\alpha<1$.","marker":"[12]"},{"why":"Provides the function-space facts, including Sobolev and Besov embeddings and the smooth local theory, used throughout Sections 2 and 5.","marker":"[2]"},{"why":"The Kato–Ponce commutator estimate used in Lemma 5.2 to control the projected nonlinearity in the stability argument.","marker":"[10]"},{"why":"The first Calderón commutator estimate used to bound $[R_aR_b,v_c]\\partial_c F$ in Lemma 5.2.","marker":"[5]"},{"why":"The Coifman–Meyer commutator estimates that handle the singular-integral commutator in the same stability lemma.","marker":"[7]"},{"why":"Underlies the $L^j$ boundedness of Riesz transforms and the fractional-derivative interpolation used in Lemma 3.9.","marker":"[14]"},{"why":"Provides the interpolation inequalities used in the Sobolev lower bound and in the real interpolation estimates.","marker":"[3]"}],"fun_headline_variants":["Vortex rings blow up tiny fluid data in any positive regularity","Small 3D fluid data can explode instantly via vortex rings","Fractional Navier-Stokes: vortex rings cause norm inflation","Same-space blow-up in 3D hypodissipative Navier-Stokes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the strict scaling gap $\\delta=1-2\\alpha+3/p-s>0$: only this gap makes the fractional dissipative residual $\\mu^{2\\alpha}/(A\\nu)=\\zeta^{-2}\\mu^{-\\delta+b(1+1/p)}$ vanish as $\\mu\\to\\infty$, and at $\\delta=0$ the construction loses both the time-scale separation and the smallness of the nonlocal error.","fun_headline_variants_meta":{"raw":{"variants":["Vortex rings blow up tiny fluid data in any positive regularity","Small 3D fluid data can explode instantly via vortex rings","Fractional Navier-Stokes: vortex rings cause norm inflation","Same-space blow-up in 3D hypodissipative Navier-Stokes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000587,"raw_usage":{"total_tokens":2775,"prompt_tokens":984,"completion_tokens":1791,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":1715}},"tokens_in":600,"tokens_out":1791,"duration_ms":12523,"temperature":1.0,"reasoning_tokens":1715,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:28:42.482760+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct check is quantitative: for the vortex-ring ansatz of Section 3, evaluate the full residual bound (3.28) by computing the ratio $\\|\\nabla^k E(t)\\|_{L^j}/(S_{k,j}A\\nu)$ as $\\mu\\to\\infty$ with the structural parameter $\\zeta$ fixed. Lemma 3.9 predicts this ratio tends to zero precisely when $\\delta>0$; if an explicit evaluation of the torus-profile integrals for any in-range $\\alpha,p,s$ showed the ratio bounded away from zero, the stability Proposition 5.1 would fail and the approximate solution would not stay close to the exact solution on $[0,t_*]$. At the endpoint $\\delta=0$ the same calculation gives $\\mu^{2\\alpha}/(A\\nu)=\\zeta^{-2}$, matching the obstruction recorded in Remark 1.2.","supporting_citations":[{"cited_title":"Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces","cited_arxiv_id":"2504.08288","evidence_quote":"Develops the Besov picture and the sharp same-space inflation statement that the present theorem extends to $0<\\alpha<1$."},{"cited_title":"Bahouri, J.-Y","cited_arxiv_id":null,"evidence_quote":"Provides the function-space facts, including Sobolev and Besov embeddings and the smooth local theory, used throughout Sections 2 and 5."},{"cited_title":"Kato and G","cited_arxiv_id":null,"evidence_quote":"The Kato–Ponce commutator estimate used in Lemma 5.2 to control the projected nonlinearity in the stability argument."},{"cited_title":"Calderón, Commutators of singular integral operators,Proc","cited_arxiv_id":null,"evidence_quote":"The first Calderón commutator estimate used to bound $[R_aR_b,v_c]\\partial_c F$ in Lemma 5.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Coifman–Meyer commutator estimates that handle the singular-integral commutator in the same stability lemma."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underlies the $L^j$ boundedness of Riesz transforms and the fractional-derivative interpolation used in Lemma 3.9."},{"cited_title":"Bergh and J","cited_arxiv_id":null,"evidence_quote":"Provides the interpolation inequalities used in the Sobolev lower bound and in the real interpolation estimates."}],"review_version":2}