{"id":"f6e97c90-cb3d-49e8-b8a0-47960f3ddc7d","arxiv_id":"2603.03666","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Unconditional uniqueness of mild Navier-Stokes solutions fails in every Besov space with negative regularity index, via non-L^2 stationary singular solutions built by convex integration.","lead":"This paper proves that the Navier-Stokes equations can have two different mild solutions starting from the same tiny, rough initial data lying in any Besov space with negative smoothness. The proof constructs infinitely many stationary singular solutions by convex integration, and also proves a matching uniqueness result in an endpoint critical space.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 1.6 / Theorem 1.5(ii) rest on an unproved non-zeroness step: the L^2 iteration lacks the Fourier-support separation that guaranteed u≠0 in the Besov case.","rationale":"The main Besov-space theorem (Theorem 1.1) is well supported: the convex-integration construction in Proposition 4.1 gives Fourier support separation, the iteration in Section 5.1.1 verifies the paraproduct well-definedness and the stationary equation, and the passage to two mild solutions through Proposition 1.4 and Proposition 2.1 is coherent. The extended notion of mild solution via paraproduct is explicitly defined and is the standard way to interpret products of distributions in negative Besov spaces, so I do not regard that as a fatal objection; it is a caveat the authors acknowledge. The genuine soft spot is the L^2 iteration for 0<α<(d+1)/4: Proposition 4.2 provides L^2 estimates and Reynolds-stress decay, but no frequency separation, so the proof that the limiting stationary solution is nonzero is missing. The text merely says 'one can mimic' the earlier argument, but the earlier argument's non-zeroness relied on exactly the frequency separation that Proposition 4.2 lacks. Because Theorem 1.5(ii) and Corollary 1.6 are part of the paper's claimed results, this gap warrants a conditional verdict. The reader's rationale identified this step, even though their stated weakest assumption was the extended solution notion; hence partial agreement. The proposed test—adding a frequency-localization corrector or proving a projection estimate—would settle whether the L^2 iteration can be repaired without changing the main Besov result.","tokens_in":27540,"tokens_out":30812,"duration_ms":265301,"concrete_test":"Add a frequency-localization corrector to Proposition 4.2 (as in Proposition 4.1) and rerun the iteration, or prove a projection estimate: for fixed K≥1, show \\|P_{\\le K}(u-u_0)\\|_{L^2} \\to 0 as the iteration parameters \\gamma,\\mu are sent to infinity. If this estimate holds, non-zeroness of u follows from u_0≠0. A more direct check: compute \\langle w, u_0\\rangle for the first perturbation w in the L^2 iteration, using the explicit Mikado flows; show it is o(1) as the concentration and oscillation parameters grow. If the inner product is not o(1), the limit may lose the initial mode and Corollary 1.6 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.2, unlike Proposition 4.1, includes no frequency-localization conclusion: the perturbation w = w^(p)+w^(c) is not asserted to have Fourier support in a high annulus. In the proof of Theorem 1.5(ii) (Section 5.1.2), after obtaining an L^2 Cauchy sequence and a limiting singular solution u, the text says 'One can mimic the above proof' to get infinitely many solutions. The mimicked proof in Section 5.1.1 crucially used Proposition 4.1's Fourier support separation to show \\hat u(m) = \\hat u_0(m) for |m|=1, whence u ≠ 0. Without an analogous statement for the L^2 iteration, the sequence could converge to zero; then the constructed 'non-trivial' stationary solution is trivial, and the non-uniqueness Corollary 1.6 collapses. No lower bound on any Fourier mode of the limit u is given, nor an alternative reason that the projection of u onto the initial mode does not vanish. This is the under-derived non-zeroness step flagged in the reader's rationale, and it affects the L^2/small-α results specifically.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves failure of unconditional uniqueness of mild solutions to the (fractional) Navier-Stokes equations in every negative-regularity Besov space B^{-θ}_{q,r} on the torus, for any θ>0 and d≥2. The strategy is to construct non-trivial stationary singular solutions via convex integration with intermittent Mikado flows and Fourier localization. The central Besov construction (Propositions 4.1, 4.2 and Sections 6–8) is written in considerable detail: a sequence of approximate solutions (u_n, R_n) to the stationary Navier-Stokes-Reynolds system is produced, with the velocity perturbation supported in a high-frequency annulus separated from the previous frequencies, giving both convergence in all negative Besov spaces and a nontrivial limit. The paper also states L^2-based existence results for small α (Theorem 1.5(ii), Corollary 1.6) and a uniqueness theorem for stationary weak solutions in an endpoint critical space (Theorem 1.7). The Besov iteration is largely coherent, but several load-bearing steps in the L^2 claims and in the proof that the constructed data are non-smooth are missing or rely on inapplicable results.","tokens_in":27864,"tokens_out":24554,"duration_ms":194202,"significance":"If the main theorem (Theorem 1.1) is fully correct, it is a major advance: unconditional uniqueness of mild solutions had been known in subcritical Lebesgue spaces and in critical/supercritical Besov spaces only for classical solution classes, and this would be the first non-uniqueness result in subcritical negative-regularity Besov spaces. The method is original in combining convex integration with frequency localization and intermittency, and the Besov part of the paper is detailed and reproducible, with explicit estimates. The uniqueness theorem (Theorem 1.7) is also a worthwhile contribution. However, the paper's scope is limited by its definition of mild solutions via a paraproduct in H^{-s} rather than a classical L^1 tensor product, and the L^2/second-solution results as written are not fully justified.","major_comments":[{"comment":"The proof of Theorem 1.5(ii) does not establish that the limiting L^2 solution u is nonzero. Proposition 4.2 contains no Fourier-support separation: the perturbation in Section 9, w = Σ a_k W_k(γx) + ..., has frequencies that are not confined to a high annulus, so the low modes of u0 can be altered. The sentence 'One can mimic the above proof' (end of §5.1.2) is insufficient, because the Besov proof in §5.1.1 used the Fourier separation to show \\hat u(m)=\\hat u_0(m) for |m|=1, and the L^1 bound ∥u-u0∥≤1; neither is available here. Without a lower bound on some Fourier mode of the limit (or an alternative argument), the sequence could converge to zero, in which case the constructed 'non-trivial' stationary solution is trivial and Corollary 1.6 collapses.","section":"§5.1.2, Proposition 4.2"},{"comment":"The argument 'u_in /∈ C∞. Otherwise, Corollary 3.1 implies that u_in = 0' is not valid for the data constructed in that proof. Corollary 3.1 requires u∈L^p with p in the subcritical range (e.g. p>d/(2α-1) for α≤(d+2)/4). However, the iteration in Theorem 1.2(i) uses p_n=3/2, so the limit u_in is only guaranteed to lie in L^{3/2}. For α=1 and d≥2, 3/2 is strictly below the critical exponent d/(2α-1)=d, so Corollary 3.1 does not apply. The conclusion that u_in is non-smooth needs a different proof; for example, the direct energy identity for a smooth stationary solution, ∥(-∆)^{α/2}u∥_{L^2}^2 = ∫(u⊗u):∇u = 0, shows any smooth mean-zero stationary solution is zero. This is a local fix, but it must be made explicitly.","section":"§5.2, proof of Theorem 1.2(i)"},{"comment":"Corollary 1.6 is stated without proof and does not follow from Theorem 1.5(ii) alone. For 0<α<(d+1)/4, the L^2-critical condition α>(d+2)/4 fails, so the local well-posedness result of Proposition 2.1 is not available in L^2 (or in any L^p with p≤2, since d/(2α-1)>2). The stationary solution U(t)=u_in provides only one weak solution; the existence of a second weak solution with the same L^2 initial data is not established anywhere in the paper. Either a second solution must be constructed (e.g., via the same convex integration) or the corollary must be removed or qualified.","section":"Corollary 1.6"}],"minor_comments":[{"comment":"The reduction to a smaller θ near the start of §5.2 is written in a confusing manner. The embedding B^{-θ}_{q,1} ↪ B^{-θ_1}_{q_1,r_1} holds for θ_1≥θ, so to cover all θ>0 one must construct solutions for θ arbitrarily small; the text should state this clearly.","section":"§5.2, embedding reduction"},{"comment":"The inequality ∥u0∥_{B^{-θ}_{∞,1}} ≤ ∥u0∥_{L∞} in the proof of Theorem 1.2(i) is missing a constant depending on θ (since ∑_{N≥1} N^{-θ} is finite but not equal to 1). This is harmless—one can rescale u0—but the estimate as written is not exact.","section":"§5.2, choice of u0"},{"comment":"The paper correctly emphasizes that u⊗u is defined via paraproduct in H^{-s} and that the solutions may not be classical mild solutions with locally integrable nonlinearity. This point should also be reflected in the abstract and introduction, as it is essential for interpreting Theorem 1.1.","section":"§1.3, Definition 1.3"},{"comment":"There is a small formatting/typo issue: 'extended it to Tϵ∈(0,1) L2−ϵ∩ ˙H−ϵ(T2)' is unclear; the intended expression should be typeset properly.","section":"§1.3, line after Theorem 1.5"}],"recommendation":"major_revision","confidential_remarks":"The main Besov theorem appears plausible and the iteration in Sections 6–8 is detailed, but the proof contains an invalid appeal to Corollary 3.1 for non-smoothness (easily fixable) and the L^2 results (Theorem 1.5(ii), Corollary 1.6) are underderived. I recommend major revision: the authors should repair the non-zeroness step in the L^2 iteration, supply a valid proof that the constructed data are non-smooth, and either prove Corollary 1.6 or remove it. The paper would then be a strong contribution to the field."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main takeaway: the central result—failure of unconditional uniqueness in all negative-regularity Besov spaces, including subcritical ones—is new and, as far as I can tell, correct. The proof is a substantial convex-integration construction, and the authors earn their claim. I'd send this to a serious referee on the strength of Theorem 1.1 alone.\n\nThe genuinely new part is the extension of non-uniqueness to every Besov space B^{-θ}_{q,r}, θ>0, and to fractional Navier-Stokes with arbitrarily large dissipation. The construction of stationary singular solutions via a frequency-localized iteration is clean; Proposition 4.1's Fourier-support separation is what makes the nonzero limit work, and the estimates in Sections 7–8 are carefully done. The endpoint uniqueness result (Theorem 1.7) is a nice complement.\n\nNow the soft spots. The L^2 results—Theorem 1.5(ii) and Corollary 1.6, relying on Proposition 4.2—have a hole. In the Besov case, the Fourier localization gives you \\hat u(m)=\\hat u_0(m) for |m|=1, which forces the limit to be nonzero. In the L^2 iteration, Proposition 4.2 has no frequency-localization conclusion, and the proof just says \"one can mimic the above proof.\" Mimicking the Besov argument doesn't work without an analogue of the low-mode preservation. The constructed sequence could in principle converge to zero, and then the \"non-trivial\" stationary solution would be trivial. I did not find any argument in the text ruling this out. This is not a fatal flaw for the paper's main message, but the L^2 claims are not fully supported as written.\n\nThe other thing to keep in mind is the solution concept. The authors define mild solutions using a paraproduct tensor product, valid for distributions even when u⊗u is not locally integrable. They flag this in a footnote and call such solutions \"singular mild solutions.\" That's an honest choice, but it means Theorem 1.1 is a statement about this extended class, not about classical mild solutions with u⊗u ∈ L^1_loc. If the intended audience cares about the classical notion, the title overreaches slightly.\n\nOverall: recommended for peer review. The main theorem deserves a serious referee; the L^2 section needs a repaired non-zeroness argument and possibly a small rewrite. I'd want to see that fixed before accepting the whole package.","headline":"Main theorem (non-uniqueness in every negative-regularity Besov space) is a real advance and mostly well-proved; the L^2/small-α results have an unproved non-zeroness step, and the 'mild solution' convention is worth remembering.","tokens_in":28328,"tokens_out":3236,"would_cite":true,"duration_ms":30242,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A02","35K55","35Q30","42B37","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Unconditional uniqueness of Navier-Stokes mild solutions fails in every Besov space with negative regularity index, even subcritical ones, by constructing non-trivial stationary singular solutions via convex integration.","keywords":["Navier-Stokes equations","unconditional uniqueness","mild solutions","Besov spaces","convex integration","stationary singular solutions","fractional Navier-Stokes equations","paraproduct"],"falsifier":"Take one of the constructed stationary singular solutions u and compute the classical (distributional) tensor product u⊗u in D'(T^d). If u⊗u actually belongs to L^1_loc, then the paraproduct definition would coincide with the classical product and the non-uniqueness would survive the stricter classical notion. If, on the other hand, u⊗u is not locally integrable, then the claim 'two mild solutions in C([0,T];B^{-θ}_{q,r})' holds only under the paper's extended definition; verifying which of these occurs for an explicit seed (e.g., a simple trigonometric polynomial) would settle whether the res","tokens_in":27449,"feed_emoji":"🌊","tokens_out":2761,"duration_ms":28005,"temperature":0.7,"pith_summary":"The paper proves that for the Navier-Stokes equations on the torus, unconditional uniqueness of mild solutions fails in all Besov spaces with negative regularity index, including subcritical spaces where local well-posedness holds. For any θ>0 and q,r∈[1,∞], one can find arbitrarily small initial data in B^{-θ}_{q,r} that admit two distinct mild solutions in C([0,T];B^{-θ}_{q,r}). The construction relies on building non-trivial stationary singular solutions—time-independent solutions that solve the stationary equation with the nonlinear term defined as a paraproduct—and then converting them into mild solutions. The paper extends the result to fractional Navier-Stokes equations with arbitrarily large powers of the Laplacian, and proves the opposite side: uniqueness of stationary weak solutions in an endpoint critical space.","feed_headline":"Navier-Stokes uniqueness fails in every negative Besov space","feed_subtitle":"Two mild solutions share the same tiny initial data, even in subcritical spaces.","key_machinery":"The central tool is convex integration with Mikado flows, adapted to the stationary Navier-Stokes-Reynolds system div(u⊗u)+(-Δ)^α u+∇p=div R. Each iteration adds a high-frequency, highly concentrated velocity perturbation w whose Fourier support lies in a thin annular shell far above the previous frequencies. The nonlinearity u⊗u is defined not as a classical pointwise tensor product but as a paraproduct in H^{-s} for distributions modulo constants, because the constructed solutions need not lie in L^2_loc. A key arithmetic lemma ensures that the oscillatory factors σ_kk^⊥ all land in the same dyadic shell, so the perturbation stays divergence-free and its Fourier support is controlled. A fi","core_discovery":"The central claim is Theorem 1.1: for every dimension d≥2, every θ>0, and every q,r∈[1,∞], there exists divergence-free initial data u_in ∈ B^{-θ}_{q,r} with arbitrarily small norm such that two mild solutions u and v, both in C([0,T];B^{-θ}_{q,r}), satisfy u(0)=v(0)=u_in but u(t)≠v(t) for every t∈(0,T]. The proof constructs non-trivial 'singular solutions' to the stationary fractional Navier-Stokes equations via convex integration. These are distributions u with zero mean and zero divergence that satisfy the stationary equation with u⊗u defined as a paraproduct in a negative Sobolev space H^{-s}; such a u automatically gives a time-independent mild solution. Starting from a nonzero smooth s","pith_inferences":["The paper's non-uniqueness is tied to an extended notion of mild solution where u⊗u is defined as a paraproduct; if one insisted on classical mild solutions with u⊗u ∈ L^1_loc, the stationary singular solutions would not qualify, so the result would not contradict the classical Fabes-Jones-Rivière uniqueness in L^p.","The stationary singular solutions are time-independent, so they represent eternal 'background' flows that coexist with smooth solutions; this suggests that in negative-regularity data, the mere specification of the initial datum does not determine the evolution unless one also fixes the product law for distributions.","One can test the construction's robustness by asking whether the same iteration works for the classical Navier-Stokes equations in L^2-based spaces; the Fourier-shell separation and paraproduct estimates suggest that any such attempt would need a genuinely L^2-integrable product, which the current non-L^2 solutions fail.","The endpoint uniqueness result (Theorem 1.7) implies that the stationary non-uniqueness phenomenon is confined to spaces more singular than the critical line; identifying the exact critical/marginal regularity where non-trivial stationary solutions first appear is a natural next step."],"forward_implications":["If the central claim is correct, the Navier-Stokes Cauchy problem is locally well-posed but not unconditionally well-posed in every subcritical Besov space with negative regularity index.","The failure of unconditional uniqueness extends to fractional Navier-Stokes equations with arbitrarily large α, in both Besov and (for α>(d+2)/4) Lebesgue spaces L^p with p<2.","For 0<α<(d+1)/4, the construction yields non-uniqueness of weak solutions in L^2 with arbitrarily small initial data, giving infinitely many distinct weak solutions.","The uniqueness half of the paper shows that stationary weak solutions in the endpoint critical space B^{-1}_{∞,1} (or its fractional analogue) must be trivial, which contrasts with the abundance of singular stationary solutions in all negative Besov spaces.","The result sharpens the known ill-posedness boundary: below B^{-1}_{∞,∞} there are no unconditional uniqueness classes at all."],"fun_headline_variants":["Two Navier-Stokes solutions, one initial data, any negative space","Non-uniqueness spreads to all negative Besov spaces","Convex integration breaks Navier-Stokes uniqueness","Stationary singular solutions defeat uniqueness","Navier-Stokes: no uniqueness in any negative Besov space"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire construction relies on defining the nonlinear term u⊗u as a paraproduct in a negative Sobolev space H^{-s} for distributions that need not be locally integrable; if 'mild solution' is required to mean the classical integral equation with u⊗u ∈ L^1_loc, the constructed stationary singular solutions are not valid mild solutions and the non-uniqueness statement does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Two Navier-Stokes solutions, one initial data, any negative space","Non-uniqueness spreads to all negative Besov spaces","Convex integration breaks Navier-Stokes uniqueness","Stationary singular solutions defeat uniqueness","Navier-Stokes: no uniqueness in any negative Besov space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000158,"raw_usage":{"total_tokens":999,"prompt_tokens":621,"completion_tokens":378,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":299}},"tokens_in":365,"tokens_out":378,"duration_ms":3632,"temperature":1.0,"reasoning_tokens":299,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T19:04:20.486393+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take one of the constructed stationary singular solutions u and compute the classical (distributional) tensor product u⊗u in D'(T^d). If u⊗u actually belongs to L^1_loc, then the paraproduct definition would coincide with the classical product and the non-uniqueness would survive the stricter classical notion. If, on the other hand, u⊗u is not locally integrable, then the claim 'two mild solutions in C([0,T];B^{-θ}_{q,r})' holds only under the paper's extended definition; verifying which of these occurs for an explicit seed (e.g., a simple trigonometric polynomial) would settle whether the res","supporting_citations":[],"review_version":1}