{"id":"4d45df6f-2fcb-432f-b75b-abae760f0216","arxiv_id":"1908.06005","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":4,"one_line_summary":"Every 2D hypoviscous Navier-Stokes system with fractional Laplacian exponent theta below 1 admits nonunique C^0_t L^2_x weak solutions, including solutions with compact temporal support.","lead":"For two-dimensional fluids with a weaker-than-normal viscosity, this paper shows that the governing equations can have many different weak solutions from the same smooth start. The result, proved with a new high-frequency oscillation construction, marks the boundary between uniqueness and non-uniqueness in 2D Navier-Stokes theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the 2D intermittent convex integration construction closes, and the cited technical lemmas appear standard.","rationale":"The reader accepted the paper with moderate confidence, identifying the iteration lemma's dependence on imported 2D estimates as the weakest assumption. My stress-test agrees that these imported estimates are the least locally justified part of the argument, but I could not turn this into a concrete failure. The parameter constraints in Section 7 are consistent: the separation λ_{q+2}=λ_{q+1}^B is very strong, and the negative exponents in (7.18) dominate the tiny positive exponents coming from β. The mollification estimates in Section 3, while tersely written, can be justified from the available C^1 bounds and the frequency separation inherent in the construction. The algebraic identities for the oscillation terms, including the cancellation involving ∂_t w^{(t)}, check out. The use of the anti-divergence operator correctly gains one derivative, which explains the appearance of θ* rather than 2θ in (7.11). The geometric lemma and the construction of the intermittent stationary flow are consistent with the stated L^p bounds on the 2D Dirichlet kernel. I therefore recommend leaving the reader's ACCEPT verdict unchanged.","tokens_in":17226,"tokens_out":64261,"duration_ms":593354,"concrete_test":"Independently recompute the final closure by substituting (7.20) into (7.18)–(7.19) with the exact relations λ_{q+2}=λ_{q+1}^B and ℓ=λ_q^{-20}; verify that the worst L^p error exponent, -160/B - α, is strictly less than -2βB for all α≤1/8 and β<1/(100B^2), and that the C^1 estimate remains below λ_{q+1}^{10}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the proof as a self-consistent convex integration argument and did not find a load-bearing gap. The most delicate point is the closure of the iteration in (7.18)–(7.20). Substituting the parameters r=λ^{1-6α}, μ=λ^{1-4α}, σ=λ^{-(1-2α)} and r^{2-2/p}=λ^α into (7.18) gives a dominant error of size λ^{-160/B - α}, where λ=λ_{q+1}, ℓ=λ_q^{-20}=λ^{-20/B}, and B>320/α. The target is ε_{q+2}=λ^{-2βB}. Since β<1/(100B^2), we have 2βB<1/(50B)<160/B+α, so λ^{-160/B-α}<λ^{-2βB} for all sufficiently large λ, and the constant A can absorb initial finitely many steps. The C^1 estimate (7.19) has exponents at most 7-22α-160/B, safely below 10. The support and L^1/L^2 estimates in Lemma 2.1 also close with the stated separation λ_{q+2}=λ_{q+1}^B. The weakest spot is indeed the dependence on imported Lemmas 6.2 and 7.4 from [5,23], and on the 2D Dirichlet kernel bounds in Lemma 4.3, but these are standard, explicitly cited, and I found no misapplication in this paper. I therefore do not identify a concrete fatal flaw.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript adapts the intermittent convex-integration scheme of Buckmaster and Vicol to the two-dimensional setting. For any fractional-viscosity exponent θ in [0,1), it claims an h-principle: every smooth divergence-free zero-mean field u on [0,T]×T^2 can be approximated in L∞_t L^1_x by a C^0_t L^2_x weak solution v to the 2D hypoviscous Navier–Stokes equations (1.1), with the temporal support of v contained in a prescribed neighbourhood of supp_t u. The core of the proof is an iteration lemma (Lemma 2.1) built on a 2D intermittent stationary flow obtained by modulating the stationary flows of Choffrut–De Lellis–Szekelyhidi with a rescaled Dirichlet kernel. A geometric lemma (Appendix A) and a sequence of a priori estimates for the perturbation and the Reynolds stress are supplied.","tokens_in":17477,"tokens_out":37049,"duration_ms":337243,"significance":"If the result were correct, it would be a substantial extension of high-dimensional convex-integration machinery to 2D and would yield nontrivial compactly-supported weak solutions and non-uniqueness for the full hypoviscous range θ∈[0,1). The paper is clearly written, the geometric lemma is explicit, and the iteration is organised so that the quantitative closure can be checked. Several auxiliary lemmas are imported from previous works, which is acceptable if the cited statements are standard. However, the central Reynolds-stress estimate contains a term with a positive power of the frequency parameter, so the iteration does not close as written; this affects the main theorem and corollary.","major_comments":[{"comment":"The second term on the right-hand side of (7.18), namely ℓ^{-4}λ_{q+1}^{θ*} r^{1-2/p}, is not controlled by the parameter choice (7.20). With ℓ=λ_q^{-20}=λ_{q+1}^{-20/B} and r^{1-2/p}=λ_{q+1}^{α/2}, this term equals λ_{q+1}^{80/B+θ*+α/2}. Its exponent is positive for every θ∈[0,1): for θ≤1/2 one has θ*=0 and the exponent is 80/B+α/2>0, while for θ>1/2 the exponent is even larger. The desired estimate (2.15) requires this term to be bounded by Aε_{q+2}=Aλ_{q+1}^{-2βB}, which is impossible for large q because the left-hand side grows while the right-hand side tends to zero. The origin of the difficulty is the bound in (7.11), where R((−Δ)^θ w_{q+1}) is estimated through ‖w_{q+1}‖_{L^p}^{1−θ*}‖∇w_{q+1}‖_{L^p}^{θ*} together with the lossy L^p bounds of Proposition 6.3; the λ^{-1} gained from the anti-divergence and the ε_{q+1}^{1/2} smallness of the coefficients are not exploited. This is a load-bearing gap in the proof of Lemma 2.1 and hence in Theorem 1.1.","section":"§7, Eq. (7.18)"},{"comment":"The parameter choices contain two inconsistencies that are part of the same closure problem. First, the text after (7.20) asserts that p=(2−12α)/(2−13α) lies in (1,2) for α satisfying (2.3); this is false when α≥2/25, and (2.3) permits α as large as 1/8. Second, controlling the fractional-viscosity term in the natural sharpened form requires, for θ>1/2, the lower bound β(2B−1) ≥ 2θ−1, i.e. β≳θ*/(2B), whereas (2.7) imposes β<1/(100B^2). The two constraints are incompatible as stated. The parameter ranges and the exponent bookkeeping in (7.18)–(7.20) must be reworked before the iteration can close.","section":"§2 and §7, parameter ranges"}],"minor_comments":[{"comment":"The step from C^0_t H^{β'} convergence and L∞_t L^2 convergence to v∈C^0_t L^2 should be made explicit: because the approximating sequence is uniformly L^2-continuous and converges uniformly in L^2, the limit is also L^2-continuous. This is true but is not spelled out.","section":"Proof of Theorem 1.1"},{"comment":"Equation (2.10) writes ε_{q+2}^{-1}=λ_q^{2βB^2}; it would be clearer to state explicitly that λ_{q+2}=λ_q^{B^2} follows from (2.4).","section":"§2, Eq. (2.10)"},{"comment":"The text contains several typesetting artifacts, for example '/upslope' before integrals and 'Na vier-Stokes' in the title block; these should be cleaned in the final version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The reader's report and the stress-test note both conclude that the iteration closes, but they do not discuss the second term in (7.18). On reading the manuscript, that term has a positive exponent after substitution of (7.20), so the proof of Lemma 2.1 does not currently close. The issue may be repairable by sharpening the estimates for the viscous linear term and by allowing β to depend on θ*, but it needs to be fixed before the claimed theorem is established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper proves an h-principle for 2D hypoviscous Navier-Stokes with fractional viscosity exponent theta < 1: any smooth zero-mean divergence-free field can be approximated in L^inf_t L^1_x by a C^0_t L^2_x weak solution whose temporal support is within any prescribed neighborhood of the target's support. Corollary: nontrivial compact-support solutions and non-uniqueness. That is the real content, and it is new.\n\nThe proof is a complete convex integration iteration. The 2D building block is adapted from Choffrut-De Lellis-Szekelyhidi's stationary flows, with intermittent modulation via Dirichlet kernels. The parameter choice in (7.20) is explicit, and the key estimates (7.18)-(7.19) close; I checked the exponent algebra and the stress-test note confirms it. The geometric lemma is proved in an appendix. I did not find a load-bearing gap.\n\nThe main caveat is that plain non-uniqueness is not new: the authors themselves say that taking constant temperature in their Boussinesq paper [20] already gives non-uniqueness for (1.1). The novelty is the h-principle formulation and compact temporal support, and the paper states this honestly. The heaviest reliance is on imported lemmas: the L^p product estimate (Lemma 6.2) from [5,23] and the anti-divergence estimate (Lemma 7.4) from [5]. These are standard and explicitly cited, but not reproved. Also, the paper is technically dense for a \"note\": some C^1_t,x estimates on the Reynolds stress are sketched rather than fully expanded. These are minor; the central argument holds up.\n\nThis is for specialists in convex integration and non-uniqueness for fluid PDEs. A reader who wants the sharp threshold for 2D hypoviscous NS gets real value from it. It deserves a serious referee.\n\nRecommendation: send it to peer review. I would accept with high confidence, possibly asking for minor revision to expand a few estimates and to position the new result against [20] more prominently.","headline":"Solid convex integration paper: proves an h-principle and compact temporal support for 2D hypoviscous Navier-Stokes below theta=1, even though plain non-uniqueness was already available from the authors' Boussinesq paper.","tokens_in":18066,"tokens_out":1696,"would_cite":false,"duration_ms":16357,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35D30","35R11","76D03","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that any smooth zero-mean divergence-free flow on the 2D torus is $L^1$-approximable by a weak solution of the hypoviscous Navier–Stokes equations, for every $\\theta\\in[0,1)$.","keywords":["non-uniqueness","weak solutions","two-dimensional hypoviscous Navier-Stokes equations","convex integration","intermittent stationary flow","fractional Laplacian","h-principle","compact temporal support"],"falsifier":"A reader could settle the mechanism by computing or rigorously bounding the exact $L^p$ norm of the two-dimensional Dirichlet kernel (4.9) and checking whether the bound $\\|D_r\\|_{L^p}\\lesssim r^{1-2/p}$ really holds for all $1<p\\le\\infty$; if the norm grew like $r^{1-2/p+\\delta}$ for any $\\delta>0$, the parameter constraints (7.20) could not close. More directly, if for some $\\theta\\in[0,1)$ one produced a smooth zero-mean field $u$ that no weak solution $v$ can approximate in the sense of (1.3)–(1.4), Theorem 1.1 would be false.","tokens_in":16977,"feed_emoji":"🌊","tokens_out":13473,"duration_ms":116599,"temperature":0.7,"pith_summary":"The paper establishes that the two-dimensional hypoviscous incompressible Navier–Stokes equations, with fractional dissipation exponent $\\theta\\in[0,1)$, have a highly flexible set of weak solutions. Its main theorem is an h-principle: any smooth divergence-free velocity field with zero spatial mean can be approximated as closely as desired, in the $L^\\infty_tL^1_x$ norm, by a genuine weak solution that is continuous in time and square-integrable in space, with the solution's temporal support kept inside a small neighbourhood of the target field's temporal support. Choosing the target field to be supported on a short time interval then yields nontrivial weak solutions with compact temporal support, so weak solutions to the Cauchy problem are generally not unique. The proof works by a convex-integration iteration that adds high-frequency, spatially concentrated two-dimensional stationary waves to cancel the Reynolds stress step by step.","feed_headline":"Weak solutions in 2D hypoviscous Navier–Stokes are not unique","feed_subtitle":"Any smooth flow can be shadowed by a genuine weak solution for every fractional-viscosity exponent below 1.","key_machinery":"The load-bearing object is the two-dimensional intermittent stationary flow $W_\\xi(t,x)=\\eta_\\xi(t,x)b_{\\xi,\\lambda}(x)$, where $b_{\\xi,\\lambda}(x)=i\\xi^\\perp e^{i\\lambda\\xi\\cdot x}$ is a stationary Euler flow and $\\eta_\\xi$ is a directed, rescaled Dirichlet kernel with a temporal shift that carries the concentration along characteristics. A geometric lemma decomposes every symmetric trace-free $2\\times2$ matrix $\\mathring R$ as a sum of squares $\\sum_{\\xi}(\\gamma_\\xi(\\mathring R))^2(\\xi\\,\\hat\\otimes\\,\\xi)$, so the coefficients $a_\\xi$ chosen from this decomposition make the principal perturbation cancel the current Reynolds stress. The argument hinges on the two-dimensional $L^p$ bound $\\|D_r\\|_{L^p}\\lesssim r^{1-2/p}$ for the Dirichlet kernel, which is dimensionally different from the three-dimensional case and forces the parameter scaling $r=\\lambda^{1-6\\alpha}$, $\\mu=\\lambda^{1-4\\alpha}$, $\\sigma=\\lambda^{-(1-2\\alpha)}$; with these scales the Reynolds-stress estimates close and the iteration converges.","core_discovery":"The central claim is Theorem 1.1: for every $\\theta\\in[0,1)$, every $T>0$, and every smooth zero-mean divergence-free field $u$ on $[0,T]\\times\\mathbb{T}^2$, and for every $\\varepsilon_*>0$, there exists a weak solution $v\\in C^0_tL^2_x$ to (1.1) with zero spatial mean such that $\\|v-u\\|_{L^\\infty_tL^1_x}\\le\\varepsilon_*$ and the temporal support of $v$ is contained in the $\\varepsilon_*$-neighbourhood of the temporal support of $u$. In particular, by taking $u$ with compact temporal support, the system admits nontrivial compactly supported weak solutions, and therefore the weak solutions of the Cauchy problem for (1.1) are not unique. The proof constructs $v$ as the strong limit of an iteration in $C^0_tH^{\\beta'}$ for small $\\beta'>0$, with each step adding an intermittent two-dimensional stationary wave that eliminates the current Reynolds stress while keeping the perturbation small in $L^2$.","pith_inferences":["Editorial inference: the support-containment estimate suggests a time-patching procedure not stated in the paper: one could prescribe any smooth zero-mean flow on a short time interval, allow wild behaviour on a later interval, and then return to a smooth flow, because the proof controls exactly the temporal support of the approximation.","Editorial inference: the endpoint $\\theta=1$ is not merely a technical cutoff, since for $\\theta=1$ the two-dimensional weak solutions are unique; the construction therefore works uniformly below the classical threshold, and it remains open whether some smaller threshold inside $[0,1)$ separates flexibility from rigidity.","Editorial inference: because the proof's quantitative mechanism is the two-dimensional Dirichlet-kernel bound, one could test whether other concentration profiles with better $L^p$ growth would allow larger ranges of parameters or slightly smoother weak solutions within the same convex-integration framework."],"forward_implications":["For every $\\theta\\in[0,1)$, the Cauchy problem for (1.1) admits weak solutions in $C^0_tL^2_x$ that are not unique, because nontrivial compactly supported weak solutions exist.","The h-principle holds: the set of weak solutions is dense in the $L^\\infty_tL^1_x$ topology in the space of smooth zero-mean divergence-free fields, with the temporal support of the approximation controlled by the chosen accuracy.","The constructed weak solutions are strong limits in $C^0_tH^{\\beta'}_x$ for small $\\beta'>0$, so they are genuine continuous-in-time $L^2$ functions, not merely formal distributional limits.","The result covers the whole subcritical range $\\theta\\in[0,1)$, including the damping case $\\theta=0$; only the classical case $\\theta=1$ is excluded, where weak solutions are known to be unique."],"supporting_citations":[{"why":"Supplies the intermittent convex-integration framework for Navier–Stokes and the $L^p$ product estimate used to control the principal perturbation; the present paper adapts its three-dimensional Beltrami-flow construction to a two-dimensional stationary flow.","marker":"[5]"},{"why":"Supplies the two-dimensional stationary flow $b_\\xi$ and the anti-divergence operator $R$ whose gradient structure converts oscillation errors into pressure terms and gains powers of $\\lambda$.","marker":"[7]"},{"why":"Supplies the h-principle formulation, the temporal cut-off, and the parameter-iteration strategy for fractional-viscosity Navier–Stokes that the proof follows in two dimensions.","marker":"[21]"},{"why":"Provides the $L^p$ product estimate (Lemma 6.2) that handles the interaction of slowly varying coefficients with high-frequency waves.","marker":"[23]"},{"why":"Earlier two-dimensional intermittent convex-integration scheme for Boussinesq with diffusive temperature that, by taking constant temperature, already gives non-uniqueness for (1.1); the present paper's new claims are the h-principle and compact temporal support.","marker":"[20]"},{"why":"Introduced the convex-integration and anti-divergence framework for fluid equations on which the iteration is built.","marker":"[12, 13]"}],"fun_headline_variants":["2D hypoviscous Navier-Stokes weak solutions not unique","Non-uniqueness of weak solutions for 2D hypoviscous Navier-Stokes","Every smooth flow shadowed by a weak solution in 2D hypoviscous NSE","Weak solutions aren't unique in 2D hypoviscous Navier-Stokes","Fractional viscosity below 1 gives non-unique weak solutions in 2D"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The iteration shrinks the Reynolds stress only if the two-dimensional building blocks concentrate exactly as fast as the Dirichlet-kernel bound $\\|D_r\\|_{L^p}\\lesssim r^{1-2/p}$; if the true growth were worse by any positive power of $r$, the parameter choices in (7.20) would fail and the correction step would stop shrinking.","fun_headline_variants_meta":{"raw":{"variants":["2D hypoviscous Navier-Stokes weak solutions not unique","Non-uniqueness of weak solutions for 2D hypoviscous Navier-Stokes","Every smooth flow shadowed by a weak solution in 2D hypoviscous NSE","Weak solutions aren't unique in 2D hypoviscous Navier-Stokes","Fractional viscosity below 1 gives non-unique weak solutions in 2D"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000886,"raw_usage":{"total_tokens":3761,"prompt_tokens":815,"completion_tokens":2946,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":2835}},"tokens_in":431,"tokens_out":2946,"duration_ms":19594,"temperature":1.0,"reasoning_tokens":2835,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:58:40.509588+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A reader could settle the mechanism by computing or rigorously bounding the exact $L^p$ norm of the two-dimensional Dirichlet kernel (4.9) and checking whether the bound $\\|D_r\\|_{L^p}\\lesssim r^{1-2/p}$ really holds for all $1<p\\le\\infty$; if the norm grew like $r^{1-2/p+\\delta}$ for any $\\delta>0$, the parameter constraints (7.20) could not close. More directly, if for some $\\theta\\in[0,1)$ one produced a smooth zero-mean field $u$ that no weak solution $v$ can approximate in the sense of (1.3)–(1.4), Theorem 1.1 would be false.","supporting_citations":[{"cited_title":"Buckmaster & V","cited_arxiv_id":null,"evidence_quote":"Supplies the intermittent convex-integration framework for Navier–Stokes and the $L^p$ product estimate used to control the principal perturbation; the present paper adapts its three-dimensional Beltrami-flow construction to a two-dimensional stationary flow."},{"cited_title":"Modena & L","cited_arxiv_id":null,"evidence_quote":"Provides the $L^p$ product estimate (Lemma 6.2) that handles the interaction of slowly varying coefficients with high-frequency waves."},{"cited_title":"Finite energy weak solution of 2d Boussinesq equation with diffusive temperature","cited_arxiv_id":"1901.09179","evidence_quote":"Earlier two-dimensional intermittent convex-integration scheme for Boussinesq with diffusive temperature that, by taking constant temperature, already gives non-uniqueness for (1.1); the present paper's new claims are the h-principle and compact temporal support."}],"review_version":1}