{"id":"3e36e5d6-fedc-4489-b583-1ed67ecb3f83","arxiv_id":"1908.03529","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every θ<1/3, in a natural space of C^θ weak solutions of Euler, a residual set has kinetic energy in C^{2θ/(1-θ)} but in no better fractional Sobolev class, and smooth solutions are nowhere dense.","lead":"This paper proves that among rough Hölder continuous solutions of the incompressible Euler equations with regularity below the Onsager threshold 1/3, the typical solution has kinetic energy with exactly the sharp Hölder regularity, and no better Sobolev regularity on any time interval. It also shows smooth solutions are a sparse, nowhere dense subset of all such rough solutions, which is a step toward the Isett-Oh conjecture on energy profiles.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Parameter ordering in §3.2 forces η*>(θ')*, so e=∫|u|²+(ρ/2)f need not be C^{η*}; Proposition 2.2 is applied without its required energy regularity.","rationale":"The paper's central claim is credible and, in substance, likely true, but the proof as written contains an internal consistency gap in the Baire argument. The reader correctly identified the restriction to Xθ and the strict positivity of the energy profile as the main structural limitations. My concern is different: even within Xθ, the proof of Theorem 1.2 applies Proposition 2.2 to an energy profile whose Hölder regularity is not established. The text fixes parameters with θ' < θ'' < β < η, which forces η* > (θ')*; for a general u ∈ C^{θ'}, the energy e_u is only known to be C^{(θ')*}, so the C^{η*} bound used in the proof is unjustified. Since Proposition 2.2 explicitly requires e ∈ C^{η*} with a norm bound, this is not a question of external consensus but of the proof's internal logic. The fix is local: choose the ordering θ < θ'' < β < η < θ' (with η* < (θ')*), which still allows f to provide the sharp bad regularity and permits the convex-integration step. Because the gap is localized and repairable without changing the main construction, I would not reject the paper; I would make acceptance conditional on correcting the parameter ordering and verifying that all estimates in §3.2 and §5 remain valid under that ordering.","tokens_in":20778,"tokens_out":23438,"duration_ms":240699,"concrete_test":"Independently re-derive the regularity of the energy profile in §3.2 under the stated ordering. Take a C^{θ'} Euler solution (for instance produced by the same convex-integration construction) whose energy lies in C^{(θ')*} but not in C^{η*} for some η* > (θ')*, choose f ∈ C^{η*} \\ W^{η*}, and compute the C^{η*} norm of ẽ(t) = Γ²∫|u(Γt)|² + (δ₁/2)f(Γt). If this norm is infinite, Proposition 2.2 cannot be applied. Then re-run the Baire argument with the corrected ordering θ < θ'' < β < η < θ' and check that all inequalities in the claim still hold; if they do, the defect is a fixable parameter-order typo rather than a fatal flaw.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 1.2 (§3.2), after assuming θ* < (θ')* < θ* + 1/(2m), the authors fix θ < θ' < θ'' < β < η with η* < θ* + 1/(2m). Since the map x ↦ 2x/(1−x) is increasing, this implies (θ')* < η*. They then set e(t) = ∫_{T³}|u|² dx + (ρ/2)f(t), with u ∈ C^{θ'} and f ∈ C^{η*} \\ W^{η*}. For the rescaled profile they state ‖ẽ‖_{η*} ≲ ‖e_u‖_{η*} + ‖f‖_{η*} and conclude a uniform bound E, allowing application of Proposition 2.2, whose hypothesis is e ∈ C^{η*} with finite C^{η*} norm. But Isett's regularity bound only yields e_u ∈ C^{(θ')*}; because (θ')* < η*, an arbitrary C^{θ'} weak solution need not have e_u ∈ C^{η*}, and ‖e_u‖_{η*} can be infinite. Thus the profile e is only known to be C^{min(η*,(θ')*)} = C^{(θ')*}, below the regularity required by Proposition 2.2. The gap is repairable by choosing the parameters so that η ≤ θ', e.g. θ < θ'' < β < η < θ', which is compatible with β < η < 1/3 and keeps e ∈ C^{η*} while preserving the failure of W^{θ*+1/m,q_s}(I_r); but as written, the ordering θ' < θ'' < β < η cannot supply the required C^{η*} energy profile.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Hölder-continuous weak solutions of the incompressible Euler equations on the three-dimensional torus for Hölder exponents θ<1/3. Theorem 1.1 constructs, for any strictly positive energy profile e∈C^{θ*+γ}([0,T]) with θ*=2θ/(1−θ), a C^θ weak solution whose kinetic energy equals e; this drops the smoothness assumption on e used in earlier convex-integration schemes. Theorem 1.2 states that in the space Xθ obtained as the C^θ closure of more regular weak solutions, the set of solutions whose energy lies exactly in C^{θ*} and in no higher fractional Sobolev space on any open interval is residual, partially resolving a conjecture of Isett and Oh. Theorem 1.3 states that smooth solutions are nowhere dense in the space of all C^θ weak solutions. The proofs adapt the Buckmaster–De Lellis–Székelyhidi–Vicol convex-integration iteration, mollifying the non-smooth energy profile at scale ε_q, and use a Baire-category argument. An appendix proves a spacetime regularity estimate for differences of weak solutions.","tokens_in":21133,"tokens_out":16216,"duration_ms":175768,"significance":"If the proof is completed, the paper would be a significant contribution: it establishes the sharp energy-regularity exponent 2θ/(1−θ) as a typical phenomenon for Hölder weak solutions below the Onsager threshold, and it extends the admissible energy profiles in the convex-integration construction from smooth profiles to Hölder profiles. Theorem 1.3 is a clean strengthening of the known meagerness of smooth solutions. The paper is also careful to explain the obstruction to working in the full space of C^θ solutions, namely the 'θ−β gap' discussed in Section 6. The appendix reproves the needed spacetime regularity lemma rather than merely citing it, which is a useful self-contained feature. However, the proof of Theorem 1.2 currently contains a load-bearing parameter-ordering gap that must be fixed before the central claim is established.","major_comments":[{"comment":"The parameter choices in the proof of Theorem 1.2 are internally inconsistent and, as stated, fail to verify the hypothesis e∈C^{η*} needed for Proposition 2.2. After assuming θ*<(θ')*<θ*+1/(2m), the authors fix θ<θ'<θ''<β<η with η*<θ*+1/(2m). Since the map x↦2x/(1−x) is increasing on (0,1/3), this gives (θ')*<η*. The energy profile in (3.5) is e(t)=∫_{T^3}|u|^2 dx+(ρ/2)f(t), where u∈C^{θ'}; by (1.2) one only knows e_u∈C^{(θ')*}, which is weaker than C^{η*}. The bound ‖ẽ‖_{η*}≲‖e_u‖_{η*}+‖f‖_{η*} used immediately before applying Proposition 2.2 is therefore unjustified: ‖e_u‖_{η*} may be infinite. Moreover, this ordering contradicts the later assertion 'since β<θ′' used in the verification of (3.13), since θ'<θ''<β would instead give β>θ'. The repair is to choose the parameters in the order θ<θ''<β<η<θ', with η*<θ*+1/(2m). Then (θ')*>η*, so e_u∈C^{(θ')*}⊂C^{η*}, and (3.13) is satisfiable because β<θ'. This ordering is compatible with Proposition 2.2 and with the requirement v∈C^{θ''}⊂Xθ. As written, however, the proof of the claim at the heart of Theorem 1.2 is not valid.","section":"Section 3.2, Eq. (3.5) and application of Proposition 2.2"}],"minor_comments":[{"comment":"Remark 3.1 opens with 'we fixed parameters 0<β<θ′<1/3', which conflicts with the ordering θ<θ'<θ''<β<η stated in the proof of Theorem 1.2. After the parameter ordering is corrected, this remark should be updated to be consistent with the proof.","section":"Remark 3.1"},{"comment":"In the proof of Theorem 1.3, the phrase 'C^θ close' should specify the space-time Hölder norm C^θ_{x,t}; the estimate follows from Proposition A.1, but this should be stated explicitly. Also, the conclusion that a nonconstant energy profile forces v to lie outside the uniform closure of smooth solutions would benefit from a one-sentence justification.","section":"Section 3.3, proof of Theorem 1.3"},{"comment":"The space cθ of little-Hölder functions is mentioned in Section 6 but is not defined there; a brief definition or reference would improve readability.","section":"Section 6, notation"}],"recommendation":"major_revision","confidential_remarks":"The parameter-ordering gap in Section 3.2 appears to be repairable by rearranging the parameters as θ<θ''<β<η<θ' and adjusting the surrounding estimates; the rest of the convex-integration argument is standard. I would be willing to review a revised version. The remaining issue is whether the authors choose to simply repair the ordering or also improve the exposition around Remark 3.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague: this paper has two results worth keeping and one that currently doesn't close. The solid pieces are Theorem 1.1, which adapts the Buckmaster–De Lellis–Székelyhidi–Vicol convex integration scheme to energy profiles that are only C^{θ*+γ} rather than smooth, and Theorem 1.3, which shows smooth solutions are nowhere dense in the full space of C^θ weak solutions. The proof of the inductive proposition is detailed, and the appendix reproving Isett's time-regularity lemma is a useful, self-contained addition. No circularity red flags; the self-citation to [2] is backed by a proof in the appendix.\n\nThe problem is Theorem 1.2. The stress-test note is right. In §3.2 the authors choose θ < θ′ < θ″ < β < η, which forces η* > (θ′)* because x ↦ 2x/(1−x) is increasing. The energy profile e(t) = ∫|u|² dx + (ρ/2)f(t) then combines an e_u in C^{(θ′)*} (by Isett's bound) with an f in C^{η*}. Since (θ′)* < η*, the sum is only C^{(θ′)*} in general, not C^{η*}. Yet Proposition 2.2 is applied with exactly that η, and the proof claims ‖e‖_{η*} ≤ E uniformly. For an arbitrary u in the closed set C_{m,n,r,s}, e_u need not be C^{η*}; Isett–Oh examples show it can sit exactly at the lower regularity. So the hypothesis of Proposition 2.2 is not met.\n\nThis is not a cosmetic slip. The genericity argument depends entirely on prescribing this e and getting the sharp failure of Sobolev regularity. The ordering θ′ < θ″ < β < η is forced if you want output regularity above θ′ while keeping β < η, so a repair would need a different mechanism, not just a parameter tweak.\n\nWhere does that leave the paper? Theorem 1.1 and Theorem 1.3 look sound and are genuinely useful extensions. The paper is transparent about the Xθ restriction and the strict positivity of the energy. But the advertised partial resolution of Isett–Oh Conjecture 1 rests on Theorem 1.2, and that proof currently has a load-bearing gap. I would send this to a serious referee, with the clear instruction to focus on whether Theorem 1.2 can be salvaged. If it can, this is a strong paper; as it stands, it is a solid Theorem 1.1 plus an unproved genericity claim.\n\nRecommendation: accept for peer review, but expect major revision and be ready for the possibility that Theorem 1.2 is not repairable in its current form.","headline":"Theorem 1.1 and 1.3 look solid, but Theorem 1.2 as written has a genuine parameter-ordering gap in the energy regularity; the paper deserves refereeing, not desk rejection.","tokens_in":21711,"tokens_out":7067,"would_cite":true,"duration_ms":74344,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q31","35D30","76B03","26A21"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that, below the Hölder exponent $1/3$, weak solutions of the incompressible Euler equations with kinetic energy at exactly the sharp regularity $2\\theta/(1-\\theta)$ and no better Sobolev regularity are typical in a…","keywords":["incompressible Euler equations","Hölder weak solutions","kinetic energy regularity","convex integration","Baire category","residual set"],"falsifier":"Find one open ball in $X_\\theta$ in which every solution has kinetic energy belonging to $W^{\\theta^*+\\varepsilon,p}$ on some fixed open interval for some $\\varepsilon>0$ and $p\\ge1$; Theorem 1.2 asserts no such ball exists, so any such ball would refute the residuality claim.","tokens_in":20563,"feed_emoji":"🌊","tokens_out":9693,"duration_ms":100941,"temperature":0.7,"pith_summary":"Below the critical Hölder exponent $1/3$, weak solutions of the incompressible Euler equations can dissipate energy, and the kinetic energy $e_v$ of any $C^\\theta$ solution is known to satisfy $e_v\\in C^{\\theta^*}$ with $\\theta^* = 2\\theta/(1-\\theta)$. This paper proves that this bound is sharp in a strong generic sense: inside the space $X_\\theta$ obtained by closing the smoother weak solutions in the $C^\\theta$ norm, the set of solutions whose energy is in $C^{\\theta^*}$ but in no better fractional Sobolev space $W^{\\theta^*+\\varepsilon,p}$ on any open time interval is residual, hence typical in the Baire category sense. The same machinery shows that smooth solutions form a nowhere dense set in the full space of all $C^\\theta$ weak solutions. If the genericity statement could be transferred from $X_\\theta$ to the full solution space, it would resolve the outstanding part of the sharp-energy-profile conjecture; the obstruction is a technical gap in starting the convex-integration iteration from merely $C^\\theta$ data.","feed_headline":"Hölder Euler flows with maximally irregular energy are typical","feed_subtitle":"Below the energy-conservation threshold, convex integration shows the irregular energy is the generic case, not the exception.","key_machinery":"The argument is carried by the convex-integration induction for the Euler–Reynolds system, the approximate system $\\partial_t v_q + \\operatorname{div}(v_q\\otimes v_q)+\\nabla p_q = \\operatorname{div} \\check R_q$. At each step one adds a highly oscillatory, divergence-free perturbation built from Mikado flows, which are exact periodic solutions of the Euler equations with prescribed Reynolds stress, while a time mollification of the target energy profile $e$ controls the energy gap in the perturbation step. The sharp exponent $\\theta^* = 2\\theta/(1-\\theta)$ organizes the construction: the mollification scale is chosen from $e$'s $C^{\\theta^*}$ norm, and the Baire-category argument writes the complement of $Y_\\theta$ as a countable union of closed sets $C_{m,n,r,s}$, each shown to have empty interior by perturbing any candidate ball with a solution carrying an energy profile that violates the better-Sobolev condition.","core_discovery":"The central discovery is that the sharp energy-regularity threshold is not a feature of carefully engineered counterexamples but a generic phenomenon. For every $\\theta\\in(0,1/3)$, the set $Y_\\theta$ of solutions in $X_\\theta$ whose kinetic energy $e_v$ belongs to $C^{\\theta^*}([0,T])$ but fails to belong to $W^{\\theta^*+\\varepsilon,p}(I)$ for any $\\varepsilon>0$, any $p\\ge1$, and any open interval $I\\subset[0,T]$ is residual in $X_\\theta$. Equivalently, outside a meager set, the kinetic energy has exactly the maximal Hölder regularity permitted by the a priori bound and no extra fractional differentiability on any time interval. A companion result shows that every strictly positive profile $e\\in C^{\\theta^*+\\gamma}([0,T])$ is realized by some $C^\\theta$ weak solution, extending the earlier smooth-profile construction. Finally, the paper proves that smooth solutions are nowhere dense in the full space of $C^\\theta$ weak solutions, so regularity is exceptional rather than typical.","pith_inferences":["If one could prove that every $C^\\theta$ weak solution lies in the $C^\\theta$-closure of smoother weak solutions, then $X_\\theta$ would be the full solution space and the residuality statement would settle the original conjecture completely; Section 6 of the paper suggests the current obstruction is technical rather than a genuine counterexample.","The strict positivity hypothesis is a real boundary of the method: prescribing an energy profile that vanishes on an interval would require a new mechanism, since the inductive estimates force $e(t)\\ge \\delta_1\\lambda_0^{-\\alpha}>0$.","The same Baire framework could be tested for other critical exponents, such as Besov or $L^p$-based regularity scales at the energy-conservation threshold, since the appendix already converts spatial closeness into space-time regularity at the matching exponents."],"forward_implications":["For every $\\theta\\in(0,1/3)$, there exist $C^\\theta$ weak solutions whose kinetic energy belongs to $C^{\\theta^*}$ but fails to lie in any $W^{\\theta^*+\\varepsilon,p}$ on any open interval; in particular the energy is not of bounded variation on any time interval.","Within $X_\\theta$, such maximally irregular energy behavior is residual: almost every solution in the Baire sense has energy that cannot be improved above the sharp exponent.","Smooth solutions are nowhere dense in the space of all $C^\\theta$ weak solutions, so at Hölder regularity below $1/3$ the smooth or even more regular solutions form a topologically negligible set.","Every strictly positive $C^{\\theta^*+\\gamma}$ energy profile is exactly realized by a $C^\\theta$ weak solution, so the flexibility of convex integration extends to non-smooth, merely Hölder energy profiles."],"supporting_citations":[{"why":"Supplies the convex-integration iteration for the Euler–Reynolds system that the paper adapts from smooth to merely Hölder energy profiles.","marker":"[1]"},{"why":"Provides the space-time Hölder regularity result used to upgrade spatial $C^\\theta$ closeness to $C^\\theta_{x,t}$ closeness.","marker":"[2]"},{"why":"Establishes the energy-conservation side of the $1/3$ threshold that makes the sharp energy-regularity statement meaningful.","marker":"[3]"},{"why":"Introduces the Mikado-flow perturbations and the h-principle underlying the iterative construction.","marker":"[4]"},{"why":"Proves existence of dissipative $C^\\theta$ solutions for every $\\theta<1/3$, the class of solutions studied here.","marker":"[6]"},{"why":"Establishes the upper bound $e_v\\in C^{2\\theta/(1-\\theta)}$ that defines the sharp exponent $\\theta^*$.","marker":"[8]"},{"why":"Proves sharp energy profiles in the smaller range $\\theta<1/5$ and motivates the conjecture extended here toward $\\theta<1/3$.","marker":"[9]"},{"why":"Formulates the original residuality conjecture in the space of all $C^\\theta$ weak solutions.","marker":"[10]"}],"fun_headline_variants":["Irregular energy is generic for Hölder Euler flows","Maximal energy irregularity is typical in Hölder Euler solutions","Baire typicality: rough energy in Euler flows is the norm","Sharp energy regularity is generic, not exceptional, for Euler","Hölder Euler: typical solutions have maximally rough energy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the ambient space is $X_\\theta$, the $C^\\theta$-closure of stronger weak solutions, and that the prescribed energy profile is strictly positive; if either fails, the typicality statement as proved may no longer hold.","fun_headline_variants_meta":{"raw":{"variants":["Irregular energy is generic for Hölder Euler flows","Maximal energy irregularity is typical in Hölder Euler solutions","Baire typicality: rough energy in Euler flows is the norm","Sharp energy regularity is generic, not exceptional, for Euler","Hölder Euler: typical solutions have maximally rough energy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000723,"raw_usage":{"total_tokens":3318,"prompt_tokens":1092,"completion_tokens":2226,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":708,"completion_tokens_details":{"reasoning_tokens":2142}},"tokens_in":708,"tokens_out":2226,"duration_ms":17078,"temperature":1.0,"reasoning_tokens":2142,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:25:15.002517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one open ball in $X_\\theta$ in which every solution has kinetic energy belonging to $W^{\\theta^*+\\varepsilon,p}$ on some fixed open interval for some $\\varepsilon>0$ and $p\\ge1$; Theorem 1.2 asserts no such ball exists, so any such ball would refute the residuality claim.","supporting_citations":[{"cited_title":"Regularity in time of H\\\"older solutions of Euler and hypodissipative Navier-Stokes equations","cited_arxiv_id":"1811.12870","evidence_quote":"Provides the space-time Hölder regularity result used to upgrade spatial $C^\\theta$ closeness to $C^\\theta_{x,t}$ closeness."},{"cited_title":"Constantin, W","cited_arxiv_id":null,"evidence_quote":"Establishes the energy-conservation side of the $1/3$ threshold that makes the sharp energy-regularity statement meaningful."},{"cited_title":"Isett and S.-J","cited_arxiv_id":null,"evidence_quote":"Proves sharp energy profiles in the smaller range $\\theta<1/5$ and motivates the conjecture extended here toward $\\theta<1/3$."},{"cited_title":"2, 725–804, DOI 10.1007/s00205-016-0973-3","cited_arxiv_id":null,"evidence_quote":"Formulates the original residuality conjecture in the space of all $C^\\theta$ weak solutions."}],"review_version":1}