{"id":"44ce8caa-8546-4f7d-a867-eb8d515e5643","arxiv_id":"2507.19257","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For forced 2D Euler flows built around Vishik's unstable vortex, the inviscid limit from Navier-Stokes is unique and radial when initial perturbations are o(ν^{a/γ}), but at the critical size ε~ν^{a/γ} there are viscous solutions converging to non-radial non-unique Euler solutions.","lead":"This paper proves that for a carefully chosen family of forced 2D fluid flows, the vanishing-viscosity limit either selects a unique smooth radial solution or can instead produce non-unique non-radial solutions, depending on how the size of the initial perturbation compares with a precise power of the viscosity. It identifies the exact error threshold at which the selection principle fails, connecting fluid non-uniqueness to predictability limits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central threshold rests on the spectral input: semisimple unstable eigenvalue with Reλ≥4/α; a Jordan block would break the bootstrap.","rationale":"The reader's weakest_assumption correctly identifies the spectral structure as the load-bearing input. I agree: the proof of Theorem 1.4 explicitly relies on the semisimple unstable eigenvalue and the semigroup estimate, and the paper acknowledges in Section 1.4.3 that semisimplicity is necessary to avoid errors growing faster than e^{τa}. The authors prove this property in Section 2 and Appendix A, and I found no specific flaw in that proof; the concern is that this is a long, perturbation-theoretic argument anchored in Vishik's construction, with less independent confirmation than other ingredients. I did identify a minor manuscript error: Section 1.3 defines s=1+2/(2+σ), which is impossible together with 0<s<min(2−3α/2,1); Section 5.1 uses the correct s=1−2/(2+σ). This is clearly typographical and does not affect the argument. The initial-layer controllability, the modified-background convergence, and the nonlinear bootstrap all appear internally consistent, and the parameter count is modest. The concrete test proposed would settle the spectral concern; if it passes, the reader's ACCEPT verdict stands, so I recommend no change.","tokens_in":53265,"tokens_out":34435,"duration_ms":324629,"concrete_test":"Independently verify the spectral input for the vortex from Proposition 1.2: discretize the truncated operator L_R^{(κ)} in the invariant subspace U_{m0} (e.g. Chebyshev collocation on the radial ODE in Lemma 2.9) and compute the unstable eigenvalue λ and its algebraic and geometric multiplicities; confirm Reλ≥4/α and that the L2→L2 semigroup growth over τ∈[0,50] is bounded by C e^{aτ} with a moderate constant, not C τ e^{aτ}. Additionally, rerun the perturbation step (Lemma 2.8(iv) and Proposition A.3) for the chosen truncation radius by checking that the resolvent difference bound (2.18) is below the threshold in Lemma A.2 for κ=0.01.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1's threshold κ_c=a/γ inherits its value from the unstable eigenvalue a=Reλ of the linearized self-similar Euler operator Lss. The entire proof of Theorem 1.4 depends on the semigroup bound ||e^{τLss}||_{L2→L2}≤C e^{aτ} (Corollary 2.10). This bound requires the unstable eigenvalue to be semisimple: with a Jordan block the semigroup grows like τ e^{aτ}, and the Duhamel forcing terms in (5.7) would produce errors that outgrow the linear mode, collapsing both the rigidity and non-uniqueness conclusions. The paper proves semisimplicity (Propositions 1.2 and 2.7) by perturbing Vishik's vortex and applying the abstract spectral perturbation theory of Appendix A; this is the least externally anchored part of the proof, since it imports the eigenvalue from Vishik's construction and relies on a chain of resolvent and perturbation estimates (Lemmas 2.8–2.11, Propositions A.1–A.3). A gap in any step, e.g. the truncation Lemma 2.2 or the κ-perturbation introducing a defective eigenvalue, would invalidate the central claim. This is a genuine load-bearing assumption, not merely a technical inconvenience.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inviscid limit of forced 2D Navier-Stokes equations in the Vishik non-uniqueness scenario. The main result, Theorem 1.1, identifies a critical initial-perturbation size ε ∼ ν^{κ_c}, with κ_c = a/γ, measured in a dimensionless H^{2+s}-type norm: below this size every sequence of Navier-Stokes solutions converges to the unique radial Euler solution ū, while at this size there are initial data whose solutions admit subsequences converging to non-radial Euler solutions. The proof combines spectral analysis of the self-similar linearized Euler operator (Section 2), a detailed study of the viscosity-modified background (Section 3), an approximate controllability argument for the initial layer based on backward uniqueness (Section 4), and a nonlinear bootstrap controlling the dynamics after the initial layer (Sections 5 and 6).","tokens_in":53503,"tokens_out":22852,"duration_ms":245594,"significance":"If correct, the paper is a substantial advance: it gives the first PDE example in which the vanishing-viscosity limit acts as a sharp selection principle around a non-unique Euler solution, with a quantitative threshold derived from the unstable eigenvalue rather than fitted. The architecture is explicit: the critical exponent is κ_c = a/γ, the norm is specified in (1.13), and the mechanism is decomposed into spectral, initial-layer, and long-time bootstrap components. The main risk is the load-bearing spectral input — a semisimple unstable eigenvalue with Re λ ≥ 4/α and uniform resolvent and semigroup bounds (Propositions 1.2 and 2.7, Corollary 2.10). The manuscript supplies a proof of this input through the perturbation argument in Section 2.3 and Appendix A, and I did not find a gap in that chain. Reliance on Vishik's construction and on the authors' prior works is clearly cited, and those inputs are proved elsewhere.","major_comments":[],"minor_comments":[{"comment":"The parenthetical '(νe^{γτ}≫1)' appears to have the wrong sign: with τ = log t the viscous term is ν e^{-γτ} ΔU, so the viscosity is negligible for t ≫ T_ν, where ν e^{-γτ} ≪ 1. Please correct this.","section":"Section 1.2, around (1.11)"},{"comment":"The choices of σ > 0 and ζ ∈ (0, γ) are asserted without proof. Since the explicit rate ζ(α,p) in Lemma 3.3 is continuous in p, a short justification (and the observation that |ζ(α,2)| < γ for α ∈ (0,1) when needed) would make the restriction transparent.","section":"Remark 3.4"},{"comment":"The statement repeats the quantifier block 'For any 0 < c < 1, there exist τ0 ... and C ...' twice; the duplicated sentence should be removed.","section":"Proposition 5.3"},{"comment":"The argument introduces τ0 as 'given by Theorem 1.4' before the value of c in Theorem 1.4 is chosen. Since τ0 depends on c, the proof should first fix c = 1/(16||η||_{W^{2,2+σ}}) or the desired small constant, and only then take the corresponding τ0 and C.","section":"Section 6, Step 1"},{"comment":"The statement says p ∈ (2,+∞) and later specifies p ∈ (2, 2/α) only after the theorem. The two formulations should be harmonized in the statement.","section":"Theorem 1.1"},{"comment":"Because Section 1.4.3 states explicitly that a Jordan block would invalidate the bootstrap, it would be helpful to add a sentence in the statement of Theorem 1.4 pointing to Propositions 2.7 and A.1 as the proof that the perturbed eigenvalue remains semisimple. This is a clarity request rather than a correction.","section":"Corollary 2.10 and Section 1.4.3"}],"recommendation":"minor_revision","confidential_remarks":"The paper is heavily dependent on Vishik's construction and on the authors' prior works (ABC+24, AC23, ABC22). That dependence is appropriate and transparent, but independent verification of the ultimate spectral input still traces to those sources; the present manuscript's genuinely new contributions are the semisimplicity and velocity-formulation analysis in Section 2 and the threshold construction built on it. The central claim appears sound and the exposition is generally careful, so I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should read this one. It is the first rigorous instance where vanishing viscosity preserves the non-uniqueness of forced 2D Euler solutions. The main result is sharp: initial data of size o(ν^{κ_c}) are rigidly pulled to the radial solution, while data of size c0 ν^{κ_c} can produce non-radial limits. The critical exponent κ_c = a/γ is derived from the spectral growth rate and the viscous scaling, not fitted. That is a genuine new theorem about selection principles, not a repackaging of Vishik.\n\nWhat the paper does well: it builds a detailed, coherent framework for singular perturbations of unstable self-similar solutions. The initial-layer analysis via approximate controllability is a real technical contribution, and the long-time nonlinear bootstrap is handled with care. The spectral section is the least inviting part, but the uniform resolvent estimates and the semisimplicity argument are addressed head-on. I checked the structure of the bootstrap and the Duhamel terms in Section 5; the semisimple eigenvalue is indeed load-bearing, but the paper proves it (Propositions 1.2 and 2.7, Corollary 2.10) rather than assuming it. The convergence eU → ¯U in W^{2,2+σ} is also established with a clever cancellation, and the paper is honest about where that convergence fails.\n\nSoft spots, in proportion: the proof is enormous, and the most delicate steps (truncation stability, spectral perturbation of Vishik's vortex) rest on a chain of estimates that would take a specialist weeks to fully verify. I did not find a concrete error, but the correctness confidence should be moderate rather than high. The result holds for a carefully chosen force and a specially tailored norm; it does not touch convex-integration Euler solutions, which remain open. The reliance on Vishik's background vortex is explicit and unavoidable, so it is a limitation of scope, not a hidden assumption. Minor typos exist, but nothing that undermines the argument.\n\nWho is this for: anyone working on Euler/NS selection, spontaneous stochasticity, or instability in fluid PDEs. It deserves serious refereeing. I would send it out, and I would cite it if I worked in this area.","headline":"A serious, carefully proved answer to whether vanishing viscosity selects a unique Euler solution in Vishik's forced 2D scenario: it does below a precise threshold and fails at that threshold.","tokens_in":54048,"tokens_out":968,"would_cite":true,"duration_ms":12972,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q31","35Q30","76B03","76D05","35B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Vanishing viscosity selects a unique radial Euler solution below a sharp initial-data threshold, and fails at the threshold.","keywords":["vanishing viscosity limit","2D Euler equations","Navier-Stokes equations","non-uniqueness","selection principle","self-similar instability","critical threshold"],"falsifier":"Compute the spectrum of the linearized self-similar operator for the truncated vortex used in Proposition 1.2; if the unstable eigenvalue is defective (has a Jordan block) or has real part below $4/\\alpha$, the bootstrap in Theorem 1.4 fails and the dichotomy cannot hold. A numerical check of the theorem would be to simulate the forced Navier-Stokes equations at $\\varepsilon=c_0\\nu^{\\kappa_c}$ and test whether the non-radial component grows approximately as $t^a$; if it does not, the threshold claim is false.","tokens_in":53032,"feed_emoji":"🌊","tokens_out":10140,"duration_ms":91098,"temperature":0.7,"pith_summary":"This paper proves that, for a specially chosen body force in the two-dimensional incompressible Euler equations, the vanishing-viscosity limit is a selection principle only below a sharp threshold in the size of the initial perturbation. The threshold is $\\varepsilon \\sim \\nu^{\\kappa_c}$ with $\\kappa_c = a/\\gamma$, where $a$ is the growth rate of the unstable mode of a self-similar background vortex and $\\gamma = 2/\\alpha - 1$ sets the viscous crossover time. If the perturbation is much smaller than the threshold, every Navier-Stokes solution converges to the unique radial Euler solution; if the perturbation is exactly at the threshold, some sequences of viscous solutions converge instead to non-radial Euler solutions different from the radial one. The result thereby turns the known non-uniqueness of forced 2D Euler into a quantitative statement about predictability of the inviscid limit.","feed_headline":"A sharp threshold decides when viscosity selects one Euler solution","feed_subtitle":"Below ν^κ_c every viscous solution converges to the radial Euler flow; at the threshold, non-radial limits appear.","key_machinery":"The argument is carried by the spectral structure of a self-similar unstable background vortex. The key object is the linearized self-similar Euler operator $L_{\\rm ss}$ defined in (1.10), acting on $m_0$-fold rotationally symmetric divergence-free vector fields; it has an unstable eigenvalue $\\lambda$ with $a=\\operatorname{Re}\\lambda\\ge 4/\\alpha$, semisimple, whose eigenfunction $\\eta=q(r)e^{im_0\\theta}$ lies in $H^5$. Three further mechanisms use this structure. The dimensionless norm $Y_\\nu$ measures the perturbation in units set by the viscous length scale $L_\\nu=\\nu^{1/(2-\\alpha)}$, making the threshold scale-invariant. The initial-layer argument proves approximate controllability of the linearized Navier-Stokes flow around the viscosity-modified background by backward uniqueness of an adjoint equation. The long-time bootstrap controls the nonlinear perturbation by projecting onto the unstable spectral subspace, relying on the absence of Jordan blocks to prevent growth faster than $e^{\\tau a}$.","core_discovery":"The central claim is Theorem 1.1: there exists a fixed radial force $\\bar f$, a dimensionless norm $Y_\\nu$ equivalent to a rescaled $H^{2+s}$ norm, and a critical exponent $\\kappa_c>0$ such that the following dichotomy holds. When $\\|u^\\nu_0\\|_{Y_\\nu}=o(\\nu^{\\kappa_c})$, every sequence of Navier-Stokes solutions with that force converges as $\\nu\\to 0^+$ to the unique radial Euler solution $\\bar u$ in $L^\\infty_t(L^2\\cap W^{1,p})$. When $\\|u^\\nu_0\\|_{Y_\\nu}=c_0\\nu^{\\kappa_c}$ for a particular family, any initial datum within $c_1\\nu^{\\kappa_c}$ of that family has a subsequence of viscous solutions converging to a non-radial Euler solution $u_E\\neq \\bar u$. The critical exponent is $\\kappa_c = a/\\gamma$, the ratio of the unstable-mode growth rate to the relative scaling gap between viscosity and self-similarity, so the threshold is set by the instability itself rather than by the choice of norm.","pith_inferences":["Read as a deterministic analogue of spontaneous stochasticity, the threshold $\\varepsilon\\sim\\nu^{\\kappa_c}$ suggests a general predictability law for self-similarly unstable Euler backgrounds: the critical perturbation size should be the ratio of the fastest growth rate to the viscous scaling gap, a law one could test numerically in shear-layer or vortex-sheet instabilities.","The framework should transfer to other singular perturbations of self-similarly unstable flows, with $\\gamma$ replaced by the corresponding scaling exponent; this is an editorial extrapolation, since the paper itself studies the Laplacian viscosity.","A numerical experiment at the threshold should see the non-radial mode grow like $t^a$ in critical norms after the initial layer and reach order one by time $t=1$; observing a different growth rate would indicate that the spectral assumption fails in practice.","Perturbing the force instead of the initial datum produces two different inviscid limits by an elementary superposition trick; the substantive content of Theorem 1.1 is that the same phenomenon occurs with a fixed force and only initial-data uncertainty."],"forward_implications":["Below the threshold, the vanishing-viscosity limit is deterministic: every admissible Navier-Stokes solution converges to the radial Euler solution, with strong convergence in $L^\\infty_t W^{1,q}_x$ for every $q<2/\\alpha$ and weak convergence in the critical space $W^{1,2/\\alpha}$.","At the threshold, the inviscid limit is non-unique: within a $\\nu^{\\kappa_c}$-neighborhood of the special initial data, different subsequences may converge to different non-radial weak Euler solutions.","When the unstable eigenvalue is complex ($b\\neq 0$), the phase of the growing mode varies with the subsequence, so the limit solution itself can depend on the chosen sequence of viscosities.","The critical exponent quantifies the observation error needed for prediction: errors of order $\\nu^{\\kappa_c}$ sit exactly at the boundary between deterministic and non-deterministic behaviour of the inviscid limit.","The viscosity-modified background, the forced heat solution with the same force, returns to the inviscid background in $H^{2+s}$ as $\\tau\\to+\\infty$; this convergence is what lets the construction land on the inviscid unstable manifold after the viscous initial layer."],"supporting_citations":[{"why":"Supplies the non-uniqueness scenario: a self-similarly unstable vortex and the linearly growing mode used as the backbone of the construction.","marker":"[Vis18a]"},{"why":"Completes the scenario by constructing the non-unique forced Euler solutions whose vanishing-viscosity behaviour this paper tests.","marker":"[Vis18b]"},{"why":"Provides the exposition and spectral details for the unstable vortex construction that Proposition 1.2 adapts to velocity variables and to the simplicity assumption.","marker":"[ABC+24]"},{"why":"Supplies the uniform-in-mode resolvent and semigroup estimates for the linearized velocity operator around a vortex used in Section 2.","marker":"[GS19]"},{"why":"Provides the vortex truncation that makes the spectral problem amenable to perturbation while preserving the unstable eigenvalue.","marker":"[ABC22]"},{"why":"Supplies the eigenfunction regularity bootstrap used to obtain $\\eta\\in H^5$ and the $W^{4,2+\\sigma}$ integrability needed in the nonlinear argument.","marker":"[AC23]"},{"why":"Provides the backward-uniqueness theorem used to prove approximate controllability of the initial layer.","marker":"[Ghi86]"},{"why":"Supplies the tightness and compactness argument used in Lemma 6.1 to extract the Euler limit from the viscous solutions.","marker":"[Wu21]"}],"fun_headline_variants":["Viscous selection of Euler solutions breaks at ν^κ_c","Critical exponent decides when viscosity selects Euler flow","Sharp threshold: viscosity picks one Euler flow, then fails","Radial Euler flow wins below a critical viscosity exponent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing input is the spectral structure of the background vortex: the unstable eigenvalue of the linearized self-similar operator must be semisimple, meaning it has no Jordan block, its eigenfunction must lie in $H^5$, and this structure must persist under truncation and the self-similar perturbation. If a Jordan block appears, the perturbation can grow faster than the assumed $e^{\\tau a}$ and the bootstrap in Theorem 1.4 fails, taking both the rigidity and non-uniqueness conclusions with it.","fun_headline_variants_meta":{"raw":{"variants":["Viscous selection of Euler solutions breaks at ν^κ_c","Critical exponent decides when viscosity selects Euler flow","Sharp threshold: viscosity picks one Euler flow, then fails","Radial Euler flow wins below a critical viscosity exponent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000664,"raw_usage":{"total_tokens":3023,"prompt_tokens":930,"completion_tokens":2093,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":2028}},"tokens_in":546,"tokens_out":2093,"duration_ms":16689,"temperature":1.0,"reasoning_tokens":2028,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:58:09.909447+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the spectrum of the linearized self-similar operator for the truncated vortex used in Proposition 1.2; if the unstable eigenvalue is defective (has a Jordan block) or has real part below $4/\\alpha$, the bootstrap in Theorem 1.4 fails and the dichotomy cannot hold. A numerical check of the theorem would be to simulate the forced Navier-Stokes equations at $\\varepsilon=c_0\\nu^{\\kappa_c}$ and test whether the non-radial component grows approximately as $t^a$; if it does not, the threshold claim is false.","supporting_citations":[],"review_version":1}