{"id":"2c5ad821-fb8a-4a93-8117-ca68a812ad24","arxiv_id":"2506.00841","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Nontrivial stationary solutions of the 2D Navier-Stokes equations exist on the torus at regularity L^{2-epsilon} cap dot H^{-epsilon} for every epsilon in (0,1).","lead":"This paper constructs non-trivial stationary solutions to the two-dimensional Navier-Stokes equations that are slightly rougher than square-integrable. The result sharpens a recent construction and helps mark the precise regularity boundary where smooth fluid flows become flexible.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The diagonal error estimate in §4.4 requires an L2 bound on P_{>λ^2}W that Lemma 4.1(7) does not state; the Reynolds stress decay and the induction depend on this missing estimate.","rationale":"The reader's weakest_assumption identifies exactly the place where the proof as written has a citation gap that is load-bearing for the main theorem. The concern is not that the theorem is false, but that the induction step depends on an estimate that is asserted rather than proved. The fix is routine: the L2 smallness of P_{>λ^2}W follows from the rapid decay of the Fourier coefficients of the Mikado profile, but the paper does not show it. Because the gap is local and likely repairable, a conditional verdict is the right disposition, and no change to the reader's verdict is warranted.","tokens_in":16037,"tokens_out":14771,"duration_ms":129739,"concrete_test":"Independently derive ∥P_{>λ^2}(W^{k⊥}_{q+1})∥_{L2} directly from the construction in Lemma 4.1, e.g. via the two-derivative Bernstein bound ∥P_{>λ^2}f∥_{L2} ≤ λ^{-4}∥Δf∥_{L2} together with the derivative estimate in item 5. Check that this is small enough to make the RHS of (4.28) tend to zero and satisfy (4.29). If it is not derivable, show that Lemma 4.1(7) implies the needed L2 bound; otherwise the proof of §4.4 needs an explicit new estimate before the induction can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §4.4, the estimate of w^(p,2) ⊗ w^(p,3) at (4.28) reduces to the sum over k of ∥P_{>λ^2}(W^{k⊥}_{q+1})∥_{L2}, and (4.29) is then concluded by citing item 7 of Lemma 4.1. But item 7 only asserts smallness in L1 of P_{>λ^2}(W^{k⊥} ⊗ W^{k⊥}), not L2 smallness of P_{>λ^2}W itself. These are different objects: for a function with frequency scale ≈λ and mean zero, L2 smallness of the projection onto frequencies larger than λ^2 does not follow from L1 smallness of the projection of its tensor square without an additional argument. The bound (4.29) is used to prove (4.30), which in turn gives the decay of the Reynolds stress ∥R_{q+1}∥_{H^{-2}} in item 3 and the summability of the diagonal product in item 5. If (4.29) is not established, the induction step collapses; the convergence of u_q⊗u_q to the paraproduct and the passage to the limit in the weak formulation in Theorem 1.3 are therefore not justified as written. The needed L2 bound is very likely true and easy to prove from the Schwartz tails of the Mikado profile, but the manuscript does not supply it at the cited spot.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs nontrivial mean-zero weak solutions to the stationary 2D Navier-Stokes equations on the torus, belonging to the intersection of L^{2-epsilon} and dot-H^{-epsilon} for every epsilon in (0,1), with the nonlinear term interpreted through an absolutely summable paraproduct (Definitions 1.1 and 1.2). The proof is a Nash iteration: starting from smooth solutions of the Euler-Reynolds system, each velocity increment is built from intermittent Mikado flows (Lemma 4.1 and Definition 4.2), and Proposition 3.1 establishes the inductive estimates on L^p convergence, frequency separation, Reynolds stress decay in dot-H^{-2}, and summability of diagonal and off-diagonal products. Theorem 1.3 is then obtained by passing to the limit in the weak formulation. The paper also includes a remark that the endpoint case u in L^2 is not treated and would require a different method.","tokens_in":16430,"tokens_out":15558,"duration_ms":147917,"significance":"If correct, the result is a sharp extension of Lemarié-Rieusset's construction in dot-H^{-1} cap BMO^{-1}: it gives solutions at every regularity index strictly below L^2, with an explicit paraproduct interpretation of the nonlinear term. The proof is self-contained, the induction is carefully organized, and there is no fitting of parameters: the constants C and lambda_{q+1} are chosen to satisfy explicit inequalities. The authors also state honestly what their method does not reach, namely the L^2 endpoint. The main technical gap identified below is local and can be repaired within the manuscript's scope.","major_comments":[{"comment":"The estimate for w^{(p,2)} otimes w^{(p,3)} is reduced to a sum of terms of the form ||P_{>lambda^2}(W^{k⊥})||_{L^2} and then concluded by citing Lemma 4.1 item 7. Item 7, however, only asserts the L^1 decay of P_{>lambda^2}(W^{k⊥} otimes W^{k⊥}); it does not by itself imply the L^2 decay of P_{>lambda^2}(W^{k⊥}) that is used in (4.28). This step is load-bearing because (4.29) is used to obtain (4.30), which controls the decay of the Reynolds stress in item 3 and thus the convergence of u_q otimes u_q in the proof of Theorem 1.3. The desired estimate is very likely true and can be obtained from Lemma 4.1 item 5 with one derivative, or from the compact frequency support implicit in the construction of W; the authors should supply this argument at the cited spot rather than referring to item 7.","section":"§4.4, Eq. (4.28)–(4.29)"},{"comment":"The proof of Lemma 2.7 is only outlined, and the estimate for the second term in the decomposition (2.3) is asserted without the displayed inequality. Since this lemma is used for the oscillatory term (4.23), a complete proof should be included. In addition, the hypotheses on beta_lambda (the uniform L^1 bound and the limit ||P_{>lambda^2} beta_lambda||_{L^1} -> 0) should be verified explicitly in the application beta_lambda = W^{k⊥} otimes W^{k⊥}; this verification is not written in the paper.","section":"§2, Lemma 2.7"}],"minor_comments":[{"comment":"In the estimate of the low-frequency term, the Fourier support is described as |k| gtrsim lambda^2, but from the displayed decomposition the relevant lower cutoff is |k| gtrsim lambda; the convergence argument is unchanged, but the displayed cutoff should be corrected.","section":"§2, Lemma 2.6"},{"comment":"The construction of W^{k⊥} would be clearer if it stated explicitly that the Fourier support of W^{k⊥} is contained in an annulus of radius comparable to lambda_{q+1}; this would make item 7 immediate and would also clarify the estimate in (4.28).","section":"§4.1, Lemma 4.1"},{"comment":"The cross-term estimate is attributed to [4, Lemma 5.2] and [5, Remark 5.4] in a footnote, but the text should state the precise support-size and orthogonality argument being used, since these properties are not listed in Lemma 4.1.","section":"§4.4, Eq. (4.24)"},{"comment":"The equality between ||w_{m+1} otimes w_{m+1}||_{dot-H^{-2}} and ||P_{neq 0}(w_{m+1} otimes w_{m+1})||_{dot-H^{-2}} is used without comment; it holds because w_{m+1} is mean-zero, but this should be stated.","section":"§4.6, Eq. (4.36)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the central claim is plausible. The main issue is a local missing argument in §4.4; it is not a circularity and can be repaired. I recommend major revision rather than rejection, provided the authors supply the missing estimate and complete the proof of Lemma 2.7."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nHere’s my read of Ashkarian–Bhargava–Gismondi–Novack, the stationary 2D Navier–Stokes intermittency paper.\n\nWhat’s real: the paper constructs nontrivial stationary solutions in the intersection over ε of L^{2−ε} ∩ H^{−ε}, a genuine extension of Lemarié-Rieusset’s H^{-1} ∩ BMO^{-1} result. The new mechanism is the use of intermittent Mikado flows with a very high-frequency modulation (λ^6), which makes successive iterates decorrelated; the proof of convergence of u_q⊗u_q to the paraproduct is inductive and does not rely on any hidden fitting. The exposition is clear, and the frequency-support argument in §4.5 is clean.\n\nThe soft spot is in §4.4, the diagonal error estimate. In (4.28) they reduce ||w^{(p,2)}⊗w^{(p,3)}|| to a sum over ||P_{>λ²}W^{k⊥}||_{L²}, then cite Lemma 4.1(7) to get smallness. Lemma 4.1(7) is an L¹ statement about P_{>λ²}(W⊗W), not an L² statement about P_{>λ²}W. Those are different objects; the cited item does not imply the bound they need. Since (4.29) feeds into (4.30), and (4.30) gives the Reynolds-stress decay in item 3 and the summability in item 5, the induction as written rests on an unproven estimate. That said, I agree with the stress-test note: this is very likely fixable with a Schwartz-tail argument, because W has frequency scale λ, so P_{>λ²}W decays faster than any power of λ. The fix is routine, but it has to be written down.\n\nTwo smaller things. Lemma 2.7 is only outlined; the last term of (2.3) needs a sentence or two on why L¹ embeds into H̋^{-2} and the assumption handles it. Minor. And the paraproduct notion of solution is an interpretive choice, but it is explicit and the convergence of the product is proved, so I don’t see circularity.\n\nVerdict: the paper deserves a serious referee. I would send it out with a request that the authors supply the missing L² estimate and expand Lemma 2.7. If they do that, the result stands. I’d bring it to reading group, and I’d cite it once the gap is patched.","headline":"A genuinely new stationary-2D-Navier-Stokes construction that likely proves the claimed sharp regularity, but with one load-bearing estimate in the diagonal error term that is not justified as written and needs a routine fix.","tokens_in":16930,"tokens_out":2553,"would_cite":true,"duration_ms":23995,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","76D05","35D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The stationary 2D Navier-Stokes equations on the torus admit nonzero weak solutions at every Sobolev regularity strictly below $L^2$.","keywords":["stationary Navier-Stokes","2D torus","weak paraproduct solutions","intermittent Mikado flows","sharp Sobolev regularity","Nash iteration","L^{2-epsilon} solutions","convex integration"],"falsifier":"Compute directly the $L^2$ norm of $P_{>\\lambda/2}W_{q+1}^{k\\perp}$ for the Mikado flow of Lemma 4.1. If this norm fails to decay as $\\lambda\\to\\infty$, then estimates (4.28)-(4.29) and hence item 3 of the induction collapse, and the proof needs a different diagonal argument. A Fourier-side calculation on the explicit $\\rho_{q+1}$ profile would settle it.","tokens_in":15884,"feed_emoji":"🌊","tokens_out":8650,"duration_ms":74961,"temperature":0.7,"pith_summary":"On the two-dimensional torus, smooth stationary solutions of the incompressible Navier-Stokes equations must vanish, and the same rigidity was already known for solutions in the Lorentz space $L^{2,1}$. This paper establishes the opposite behavior at every regularity index strictly below $L^2$: there exist nonzero, non-constant stationary weak solutions in $\\bigcap_{\\epsilon\\in(0,1)} L^{2-\\epsilon}(\\mathbb{T}^2)\\cap \\dot{H}^{-\\epsilon}(\\mathbb{T}^2)$. Because the velocity field is not square integrable, the paper defines the product $u\\otimes u$ through an absolutely summable paraproduct in a negative Sobolev space. The proof is a Nash iteration using intermittent Mikado flows, and it shows that stationary Navier-Stokes solutions become flexible precisely where the energy identity loses its force. If correct, the result is essentially sharp, leaving only the endpoint $u\\in L^2$ open.","feed_headline":"Nontrivial steady 2D Navier-Stokes flows exist just below L^2","feed_subtitle":"A new iteration constructs singular torus solutions in every L^{2−ε} scale, just short of the trivial range.","key_machinery":"The central object is the intermittent Mikado flow $W_{q+1}^{k\\perp}$: a divergence-free, mean-zero oscillation in the direction $k^\\perp$, built from a periodic profile compressed into $\\lambda_{q+1}^{\\epsilon_\\gamma}$ thin parallelograms of width $\\lambda_{q+1}^{-1}$. Its $L^p$ norm scales like $\\lambda_{q+1}^{(\\epsilon_\\gamma-1)(1/p-1/2)}$, and it is used to produce velocity increments with a large frequency gap of order $\\lambda_{q+1}^{6}$ between the principal and corrector components. The paraproduct definition of $u\\otimes u$ (Definition 1.1), the geometric reconstruction of symmetric tensors (Lemma 2.5), and the super-exponential separation of frequency shells $\\lambda_{q+1}>2^{100}\\lambda_q$ together carry the induction.","core_discovery":"The paper's central claim is Theorem 1.3: for every $\\epsilon\\in(0,1)$ there is a nontrivial, mean-zero weak solution $u$ of the stationary 2D Navier-Stokes equations on $\\mathbb{T}^2$ that belongs to $L^{2-\\epsilon}$ and to $\\dot{H}^{-\\epsilon}$, with the product $u\\otimes u$ interpreted through an absolutely convergent paraproduct sum. The constructed solution is a limit $u=\\sum_q w_q$ of smooth iterates, where each increment $w_{q+1}$ is a high-frequency oscillation built from intermittent Mikado flows at frequency scale $\\lambda_{q+1}^{6}$. The Reynolds stress error $R_q$ is driven to zero only in $\\dot{H}^{-2}$, and the convergence of $u_q\\otimes u_q$ is justified by the paraproduct definition rather than by $L^2$ summability. The paper also notes that any smooth divergence-free vector field can be approximated to within $\\epsilon$ in $\\dot{H}^{-\\epsilon}\\cap L^{2-\\epsilon}$ by such solutions.","pith_inferences":["A natural test of the construction is to compute directly the $L^2$ norm of $P_{>\\lambda/2}W_{q+1}^{k\\perp}$ for the Mikado flows in Lemma 4.1; if this quantity does not decay, the diagonal Reynolds-stress estimate could instead be attempted through the $L^1$ decay of $P_{>\\lambda/2}(W\\otimes W)$ stated in the lemma.","The method is torus-specific because both the paraproduct and the Mikado flows rely on periodicity and on the absence of boundaries; a version on a bounded domain would require new boundary conditions for the renormalized product.","The huge frequency gap suggests the same iteration could work in settings where the Reynolds stress is controlled only in negative norms, for instance forced stationary flows or higher-dimensional analogues, although the intermittency exponents would need to be rebalanced.","One numerical check would be to truncate the construction to finitely many shells and compare the partial sums' Fourier energy decay with the rates predicted by membership in $L^{2-\\epsilon}$ and $B^{-1/2-\\epsilon}_{\\infty,\\infty}$."],"forward_implications":["Mean-zero stationary solutions in $L^{2,1}(\\mathbb{T}^2)$ are trivial, so the new solutions occupy exactly the largest $L^p$ scale that can support nontriviality: every exponent $2-\\epsilon$, but not $L^2$.","The approximating sequence $u_q$ converges in $L^{2-\\epsilon}\\cap\\dot{H}^{-\\epsilon}$ for every $\\epsilon\\in(0,1)$, and the paraproduct sum representing $u\\otimes u$ is absolutely convergent in $\\dot{H}^{-2}$.","By Remark 1.4, the construction perturbs any prescribed smooth divergence-free vector field into a nontrivial stationary solution within $\\epsilon$ in $\\dot{H}^{-\\epsilon}\\cap L^{2-\\epsilon}$.","The solutions lie in $\\bigcap_{\\epsilon>0} B^{-1/2-\\epsilon}_{\\infty,\\infty}$, so their Besov regularity is fixed at $-1/2-\\epsilon$ even though their $L^p$ integrability ranges over all $p<2$.","The Reynolds stress $R_q$ tends to zero in $\\dot{H}^{-2}$ but not necessarily in $L^1$; this weaker convergence is what allows the diagonal terms of $u\\otimes u$ to be controlled."],"supporting_citations":[{"why":"Establishes the $L^{2,1}$ triviality and the earlier $\\dot{H}^{-1}\\cap BMO^{-1}$ solutions that this paper sharpens.","marker":"[19]"},{"why":"Introduces intermittent convex integration for the Navier-Stokes equations, the style of construction adapted here.","marker":"[7]"},{"why":"Provides the Mikado flow construction that the paper modifies with intermittency.","marker":"[15]"},{"why":"Supplies the convex-integration framework and the intermittent Mikado flow lemma used in the increment construction.","marker":"[8]"},{"why":"Shows how to interpret nonlinear terms for sub-$L^2$ Navier-Stokes solutions by limiting procedures, motivating the paraproduct definition.","marker":"[13]"},{"why":"Provides the geometric lemma for reconstructing symmetric tensors from a finite set of directions, used to eliminate the Reynolds stress.","marker":"[9]"},{"why":"Gives the support-orthogonality estimate used to control cross terms in the principal component of the increment.","marker":"[4]"}],"fun_headline_variants":["Steady 2D Navier-Stokes admits singular solutions in every L^p for p<2","Non-square-integrable steady solutions to 2D Navier-Stokes exist","Intermittency builds singular steady 2D Navier-Stokes flows in sharp spaces","Just-below-L^2 steady Navier-Stokes solutions via intermittent construction","Sharp Sobolev singular solutions to steady 2D Navier-Stokes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The induction's Reynolds-stress bound rests on the unproved assertion that the high-frequency projection $P_{>\\lambda/2}W_{q+1}^{k\\perp}$ of each Mikado flow tends to zero in $L^2$ as $\\lambda\\to\\infty$; the paper proves only the weaker $L^1$ decay of $P_{>\\lambda/2}(W\\otimes W)$.","fun_headline_variants_meta":{"raw":{"variants":["Steady 2D Navier-Stokes admits singular solutions in every L^p for p<2","Non-square-integrable steady solutions to 2D Navier-Stokes exist","Intermittency builds singular steady 2D Navier-Stokes flows in sharp spaces","Just-below-L^2 steady Navier-Stokes solutions via intermittent construction","Sharp Sobolev singular solutions to steady 2D Navier-Stokes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002664,"raw_usage":{"total_tokens":10154,"prompt_tokens":905,"completion_tokens":9249,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":9137}},"tokens_in":521,"tokens_out":9249,"duration_ms":61284,"temperature":1.0,"reasoning_tokens":9137,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:59:13.385454+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute directly the $L^2$ norm of $P_{>\\lambda/2}W_{q+1}^{k\\perp}$ for the Mikado flow of Lemma 4.1. If this norm fails to decay as $\\lambda\\to\\infty$, then estimates (4.28)-(4.29) and hence item 3 of the induction collapse, and the proof needs a different diagonal argument. A Fourier-side calculation on the explicit $\\rho_{q+1}$ profile would settle it.","supporting_citations":[{"cited_title":"Daneri and L","cited_arxiv_id":null,"evidence_quote":"Provides the Mikado flow construction that the paper modifies with intermittency."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the $L^{2,1}$ triviality and the earlier $\\dot{H}^{-1}\\cap BMO^{-1}$ solutions that this paper sharpens."},{"cited_title":"Buckmaster and V","cited_arxiv_id":null,"evidence_quote":"Introduces intermittent convex integration for the Navier-Stokes equations, the style of construction adapted here."},{"cited_title":"Buckmaster and V","cited_arxiv_id":null,"evidence_quote":"Supplies the convex-integration framework and the intermittent Mikado flow lemma used in the increment construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows how to interpret nonlinear terms for sub-$L^2$ Navier-Stokes solutions by limiting procedures, motivating the paraproduct definition."},{"cited_title":"Shkoller, and V","cited_arxiv_id":null,"evidence_quote":"Provides the geometric lemma for reconstructing symmetric tensors from a finite set of directions, used to eliminate the Reynolds stress."},{"cited_title":"Beekie, T","cited_arxiv_id":null,"evidence_quote":"Gives the support-orthogonality estimate used to control cross terms in the principal component of the increment."}],"review_version":1}