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Vanishing viscosity non-unique solutions to the forced 2D Euler Equations

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Pith's one-line read Vanishing viscosity selects a unique radial Euler solution below a sharp initial-data threshold, and fails at the threshold.

desk verdict 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. read the letter →

arxiv 2507.19257 v1 pith:NY6TY3OO submitted 2025-07-25 math.AP

Dallas Albritton , Maria Colombo , Giulia Mescolini This is my paper · ORCID
classification math.AP MSC 35Q3135Q3076B0376D0535B35
keywords vanishingviscositylimit2DEulerequationsNavier-Stokesnon-uniquenessselectionprincipleself-similarinstabilitycriticalthreshold
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $e^{\tau a}$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

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).

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.

minor comments (6)
  1. [Section 1.2, around (1.11)] 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.
  2. [Remark 3.4] 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.
  3. [Proposition 5.3] The statement repeats the quantifier block 'For any 0 < c < 1, there exist τ0 ... and C ...' twice; the duplicated sentence should be removed.
  4. [Section 6, Step 1] 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.
  5. [Theorem 1.1] 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.
  6. [Corollary 2.10 and Section 1.4.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the critical exponent is derived from the spectral growth rate and the viscous scaling, with the spectral input proved in Section 2 rather than assumed as the conclusion.

full rationale

The paper's central threshold κ_c = a/γ is not fitted to data or defined in terms of the conclusion. It is derived in equation (1.12) from the spectral growth rate a = Re λ of the linearized self-similar Euler operator and the viscous scaling exponent γ = 2/α - 1, via the crossover time Tν = ν^{1/γ}. The rigidity statement then follows from the semigroup estimate ||e^{τL^(κ)}|| ≤ C e^{τa} (Corollary 2.10) applied to initial data of size o(ν^{κ_c}), while the non-uniqueness statement chooses initial data of size ε = C^{-1}ν^{a/γ}, so that the unstable linear mode is O(1) at time t = 1. These are genuinely derived consequences of the spectral structure, not restatements of the inputs. The spectral input itself — the existence of an unstable eigenvalue with Re λ ≥ 4/α and semisimplicity — is imported from Vishik's external construction, but the paper explicitly proves the version it needs in Section 2 and Appendix A: Proposition 1.2 is stated with the remark 'We prove Proposition 1.2 in Section 2 and Appendix A,' and the semisimplicity preservation under truncation and self-similar perturbation is established through Propositions 2.7, A.1, A.3 and Corollary A.4. The self-citations to [ABC+24], [ABC22], and [AC23] are to prior proofs of building blocks (exposition of Vishik, truncation, eigenfunction regularity), not to the present theorem, and the load-bearing spectral statements are reproven here. No step reduces by construction to its own input, and no fitted parameter is renamed as a prediction. Therefore the derivation chain is self-contained and non-circular, and the paper's self-referential reliance does not constitute circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The theorem is an existence result: it constructs a specific force, vortex, and norm. The main parameters α, s, σ, and the vortex profile are chosen by hand from a known class; they are not fitted to data. The central claim does not rest on any fitted numerical constants. The paper introduces no new physical entities such as particles or forces; the modified background and the norm Yν are mathematical constructs built from the given data.

free parameters (4)
  • α (self-similar exponent) = chosen in (0,1)
    Scaling exponent in (1.5)-(1.8). The theorem holds for the chosen α; p∈(2,2/α) requires α<1.
  • s, σ (regularity exponents) = s∈(0,min(2-3α/2,1)), s=1+2/(2+σ)
    Define the norm Yν and the spaces H^{2+s}, W^{2,2+σ}; chosen so that Sobolev embedding and Lemma 3.3 apply.
  • vortex profile Ū and m0 = from Theorem 2.1 (Vishik)
    The unstable background; all quantitative constants depend on it.
  • constants c0, c1 and observation time T = determined in the proof (Section 6)
    Normalize the initial data family and the observation time; they are derived, not fitted to data.
assumptions (5)
  • domain assumption Theorem 2.1 (Vishik): there exists a smooth vortex with unstable eigenvalue in the vorticity formulation.
    Invoked at the start of Section 2; Proposition 1.2 is proved by translating this to velocity formulation and preserving simplicity.
  • standard math Theorem 4.2 (Ghidaglia backward uniqueness)
    Used in Section 4 to prove the density of the linearized flow map (Theorem 1.3).
  • standard math Kato-Ponce commutator estimates
    Used in Lemma 4.3 to bound the commutators A and B.
  • domain assumption Well-posedness of forced Navier-Stokes in H^{2+s} for subcritical forces
    Assumed throughout the paper; standard in the subcritical regime and used to define the unique Navier-Stokes solutions.
  • standard math Kato spectral perturbation theory for semisimple eigenvalues
    Used in Appendix A to prove that the eigenvalue simplicity persists under the singular perturbation.

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Pith. "Pith review of Vanishing viscosity non-unique solutions to the forced 2D Euler Equations." pith.science (2026). https://pith.science/paper/NY6TY3OO

@misc{pith2026250719257,
  author       = {Pith},
  title        = {Pith review of: Vanishing viscosity non-unique solutions to the forced 2D Euler Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NY6TY3OO}},
  note         = {Machine review of arXiv:2507.19257}
}
abstract

The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $\nu$ and consider $O(\varepsilon)$ size perturbations of the initial datum. We discover a uniqueness threshold $\varepsilon \sim \nu^{\kappa_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which there are viscous solutions converging to non-unique, non-radial solutions.

Figures

Figures reproduced from arXiv: 2507.19257 by the authors.

Figure 1
Figure 1. Visualization of the transformation (1.24). u ν (x, t) (left) solves the Navier-Stokes equations with viscosity ν, and we observe it at time t = 1. w(y, s) (right) solves the Navier-Stokes equations with unit viscosity, and we observe it until time s = 1/Tν. The purple curve {|ξ| = const.} remains unchanged. where smax(ε) = ε − 1 a /O(1) and the big Oh is measured in critical spaces. Choose ε = ν a γ = T a ν , so th… view at source ↗
Figure 2
Figure 2. Schematic depiction of the temporal evolution of various solu￾tions. The horizontal axis represents ¯u. The blue curve represents an inviscid non-unique solution, which grows like t a in critical norms. The red curve represents the unperturbed viscous solution ue ν , which deviates by O(1) in critical norms but converges to ¯u. The purple curve represents a perturbed viscous solution u ν , which approximately lands … view at source ↗
Figure 3
Figure 3. Schematic depiction of the strategy for controlling the resolvent R(λ, L(κ) m in {Re λ ≥ ε}. In (A) and in the red half-plane in (B), resolvent estimates were proved in the red region by directly estimating the κ Rayleigh equation in (2.9). Above and below the box in (B), and inside the box but away from the eigenvalues, resolvent estimates were proved by perturbing the κ = 0 estimates. 3. The modified background In… view at source ↗

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Works this paper leans on

13 extracted references · 3 canonical work pages

  1. [2]

    Anomalous diffusion by fractal homogenization

    [AV23] Scott Armstrong and Vlad Vicol. Anomalous diffusion by fractal homogenization. arXiv preprint arXiv:2305.05048,

    arXiv

  2. [7]

    No anomalous dissipation in two-dimensional incompress- ible fluids

    [DRP24] Luigi De Rosa and Jaemin Park. No anomalous dissipation in two-dimensional incompress- ible fluids. arXiv preprint arXiv:2403.04668 ,

    arXiv

  3. [10]

    Implicit function theorem with continuous dependence on parameter

    [Maj] Pietro Majer. Implicit function theorem with continuous dependence on parameter. Math- Overflow. URL:https://mathoverflow.net/q/409166 (version: 2021-11-23). [Men24] Francisco Mengual. Non-uniqueness of admissible solutions for the 2D Euler equation with Lp vortex data. Comm. Math. Phys. , 405(9):Paper No. 207, 28,

    work page 2021

  4. [11]

    On vanishing diffusivity selection for the advection equation

    [MPS25] Giulia Mescolini, Jules Pitcho, and Massimo Sorella. On vanishing diffusivity selection for the advection equation. Annali di Matematica Pura ed Applicata (1923-), pages 1–21,

    work page 1923

  5. [1986]

    The L3-based strong Onsager theorem

    [GKN23] Vikram Giri, Hyunju Kwon, and Matthew Novack. The L3-based strong Onsager theorem. arXiv preprint arXiv:2305.18509 ,

    arXiv

  6. [1987]

    Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations

    [DM24] Michele Dolce and Giulia Mescolini. Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations. arXiv preprint arXiv:2411.18452 ,

    arXiv

  7. [2003]

    Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid

    [Vis18a] Misha Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I. arXiv preprint arXiv:1805.09426 ,

    arXiv

  8. [2013]

    Scale invariant bounds for the Kelvin-Helmholtz instability

    [KMW24] Konstantin Kalinin, Govind Menon, and Bian Wu. Scale invariant bounds for the Kelvin- Helmholtz instability. arXiv preprint arXiv:2410.14557 ,

    work page Pith review arXiv

Show all 13 references
  1. [2016]

    A proof of Vishik’s nonuniqueness Theorem for the forced 2D Euler equation.arXiv preprint arXiv:2404.15995,

    [CFMS24] ´Angel Castro, Daniel Faraco, Francisco Mengual, and Marcos Solera. A proof of Vishik’s nonuniqueness Theorem for the forced 2D Euler equation.arXiv preprint arXiv:2404.15995,

    arXiv

  2. [2018]

    Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid

    [Vis18b] Misha Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II. arXiv preprint arXiv:1805.09440 ,

    arXiv

  3. [2021]

    Anomalous dissipation and Euler flows

    [BSJW23] Jan Burczak, L´ aszl´ o Sz´ ekelyhidi Jr, and Bian Wu. Anomalous dissipation and Euler flows. arXiv preprint arXiv:2310.02934 ,

    arXiv

  4. [2023]

    Linear and nonlinear instability of vortex columns

    [AO23] Dallas Albritton and Wojciech O˙ za´ nski. Linear and nonlinear instability of vortex columns. arXiv preprint arXiv:2310.20674 ,

  5. [2024]

    Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

    [CFMS25] ´Angel Castro, Daniel Faraco, Francisco Mengual, and Marcos Solera. Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation. arXiv preprint arXiv:2502.10274 ,

    arXiv

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