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The paper proves that every C^1, (-a)-homogeneous, divergence-free initial velocity on the plane, for 1/3 < a < 1, admits a forward self-similar weak solution of the 2D Euler equations, with no smallness or sign restriction.

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2026-08-01 11:59 UTC pith:7JMJJ4MH

load-bearing objection A genuinely new large-data self-similar 2D Euler existence theorem with a believable proof; the main technical limitation (a>1/3) is honestly flagged. the 1 major comments →

arxiv 2607.19700 v1 pith:7JMJJ4MH submitted 2026-07-22 math.AP

Self-Similar Solutions of the Two-Dimensional Incompressible Euler Equation from Large Initial Data

Hyungjun Choi This is my paper
classification math.AP MSC 35Q3135C0676B0335D3046E30
keywords 2D incompressible Euler equationsself-similar solutionslarge initial datahomogeneous initial datahypodissipative equationsLorentz spacesweak solutionsvorticity transport
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that the two-dimensional incompressible Euler equations admit forward-in-time self-similar solutions starting from arbitrary large scale-invariant initial data. Concretely, for any C^1 divergence-free vector field u0 that is homogeneous of degree -a with 1/3 < a < 1, there is a profile U such that u(t,x)=t^{-a/(1+a)} U(x/t^{1/(1+a)}) solves Euler weakly with initial value u0. Previous constructions of self-similar Euler flows required the angular dependence to be close to a radial vortex or to satisfy a symmetry condition; here no smallness, sign, or radial-dominance assumption is imposed. The proof proceeds by taking a vanishing-dissipation limit of self-similar profiles for the hypodissipative (fractionally diffused) equations, with the key uniform control being a critical Lorentz-space bound on vorticity. If correct, the result supplies the large base states needed to pursue nonuniqueness and bifurcation scenarios for the Euler equations.

Core claim

On its own terms, the central discovery is that the self-similar Euler profile equation has a divergence-free distributional solution U for every (-a)-homogeneous C^1 divergence-free far-field datum, with U in the weak-L^{2/a} space, gradient in weak-L^{2/(1+a)}, and pressure in weak-L^{1/a}. The associated time-dependent field is a global weak solution that starts at u0 in the strong L^2_loc sense and whose vorticity attains the initial vorticity as a weak-* limit in the critical Lorentz space. The mechanism is a uniform-in-dissipation estimate on the vorticity of the approximating hypodissipative profiles: a truncation identity forces the rescaled distribution function k^{p_c-1}F(k) to be

What carries the argument

The load-bearing object is the vorticity truncation identity for the hypodissipative profile equation. For the dissipation parameter ε, the vorticity Ω_ε satisfies εΛ^{1+a}Ω_ε + (U_ε - y/(1+a))·∇Ω_ε - Ω_ε = 0. Testing against derivatives of (|Ω_ε| - k)_+ yields, for almost every level k, εD_ε(k) + (p_c-1)F_ε(k) - k m_ε(k) = 0, where F_ε(k) is the positive-part integral and m_ε(k) the superlevel measure; because the dissipation term is nonnegative, this gives d/dk(k^{p_c-1}F_ε(k)) ≤ 0. Together with the far-field limit of the profile, which fixes the endpoint of k^{p_c-1}F_ε(k) in terms of ∫_{S^1}|ω0|^{p_c}, this yields the uniform critical Lorentz bound; Biot-Savart and Lorentz-space estimat

Load-bearing premise

The proof needs the approximating viscous profiles to be smooth and to decay at infinity with specific rates; this is established only for a > 1/3, and when a ≤ 1/3 the iteration that produces those rates stalls, so the uniform vorticity bound and the main theorem would no longer follow from this argument.

What would settle it

Compute the hypodissipative profile sequence numerically for a fixed smooth angular datum at a = 1/3, monitoring sup_{ε∈(0,1]} ||Ω_ε||_{L^{p_c,∞}} and the far-field endpoint lim_{k↓0} k^{p_c-1}F_ε(k); if the uniform bound fails or the endpoint drifts from the value (2(p_c-1))^{-1}∫_{S^1}|ω0|^{p_c} dθ, then the central claim collapses at that endpoint. Equivalently, an analytic counterexample would be an angular datum for which the rescaled distribution function is not monotone.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For every 1/3 < a < 1 and every C^1 divergence-free (-a)-homogeneous u0, a forward self-similar weak solution exists; no smallness or sign condition is needed, so the data can be arbitrarily large.
  • The profile enjoys critical Lorentz regularity: U ∈ L^{q_c,∞}, ∇U ∈ L^{p_c,∞}, and P ∈ L^{1/a,∞}, with q_c = 2/a and p_c = 2/(1+a).
  • The solution has locally finite energy and u ∈ C([0,∞);L^2_loc), with vorticity uniformly bounded in the critical Lorentz space and weak-* converging to the initial vorticity as t↓0.
  • In the range 1/3 < a < 1/2, the vorticity transport equation ∂_tω + u·∇ω = 0 holds in distributions, so the constructed flow is a genuine weak solution of the vorticity formulation.
  • The construction provides a family of arbitrarily large self-similar Euler profiles, giving concrete base states for later instability or bifurcation analysis that could yield nonunique weak solutions with the same initial datum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: if the uniform critical vorticity estimate could be proven by a route not relying on the a > 1/3 smoothness of hypodissipative profiles, the same vanishing-dissipation strategy would plausibly extend the theorem down to a = 1/3; a numerical test of the ε-independent bound at a = 1/3 would indicate whether the threshold is real or an artifact of the present proof.
  • Editorial extension: since the same (-a)-homogeneous data class is used in recent constructions of forward self-similar Navier-Stokes solutions, the Euler profiles found here can serve as a comparison object in the vanishing-dissipation limit, potentially clarifying where the two equations' self-similar theories diverge.
  • Editorial extension: one can test the sharpness of the critical exponent by choosing angular data ω0 with a sharp spike and measuring the rate of weak-* convergence of ω(t) to ω0 in L^{p_c,∞}; the proof gives no rate, and any observed rate would constrain possible nonuniqueness mechanisms.
  • Editorial extension: the same truncation-identity mechanism may apply to transport equations with other fractional dissipation exponents or to generalized surface quasi-geostrophic profiles, where a uniform Lorentz bound is the natural bottleneck to check.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper constructs, for each 1/3 < a < 1 and every C^1 divergence-free (-a)-homogeneous initial vector field u0 on R^2\{0}, a forward-in-time self-similar weak solution of the two-dimensional incompressible Euler equations. The profile U is obtained as a vanishing-dissipation limit of hypodissipative profiles for epsilon Lambda^{1+a}; the central technical achievement is an epsilon-uniform critical Lorentz-space vorticity estimate in L^{p_c,infty}, p_c = 2/(1+a). The resulting solution satisfies u in C([0,infty);L^2_loc), u(0)=u0, and omega in L^infty_t L^{p_c,infty}_x with weak-* initial trace. For 1/3<a<1/2 the vorticity transport equation is also shown to hold. Theorem 2.1, providing smooth hypodissipative profiles with explicit far-field decay for a>1/3, is proved independently in the paper.

Significance. If the proof is correct, this is a significant advance in the theory of self-similar Euler flows: it removes smallness, sign, and radial-dominance assumptions for homogeneous initial data in the range a>1/3, and it produces large self-similar profiles that can serve as base states for nonuniqueness or instability scenarios. The paper is largely self-contained: the hypodissipative profile theorem is proved from scratch, and the appendix provides the required Lorentz-space and Biot-Savart estimates. The critical Lorentz estimate (Prop. 3.4) is elegant and robust, and the a>1/3 restriction is explicitly tied to a bootstrap threshold rather than hidden. However, the pressure normalization in Section 4 is incorrect as written, which affects the proof of the pressure part of the main theorem.

major comments (1)
  1. [§4, Eq. (4.5); §5, Prop. 5.2] The pressure normalization in Eq. (4.5) is incorrect as written. Taking the divergence of (1.6) gives ΔP_epsilon = -∂_i∂_j(U_epsilon,i U_epsilon,j); hence the correct formula is P_epsilon = -∑_{i,j} R_iR_j(U_epsilon,i U_epsilon,j), not P_epsilon = 2∑_{i,j} R_iR_j(U_epsilon,i U_epsilon,j). With the printed formula, P_epsilon is not the pressure associated to U_epsilon, so the 'straightforward verification' in Prop. 5.2 that a weak-* limit P solves (5.7) does not follow. This is a load-bearing point in the proof of Theorem 1.2(a). The fix is local: with the corrected sign and factor, the Lorentz bound in Prop. 4.2 is unchanged, and the passage to the limit in Prop. 5.2 goes through using strong L^1_loc convergence of U_epsilon ⊗ U_epsilon.
minor comments (2)
  1. [Throughout] Several cross-references are mislabeled: Prop. 3.4 cites 'Theorems 3.2 and 3.3' (should be Lemmas 3.2 and 3.3); Lemma 4.1 cites 'Theorem 2.4' (should be Prop. 2.4); Section 4 cites 'Theorem A.4' (should be Prop. A.4); 'Proof of Theorem 1.5' in Section 5 should be Corollary 1.5; and Definition 1.1 is called 'Theorem 1.1' in the statement of Theorem 1.2. These should be corrected.
  2. [§2.1.2] The construction of the Galerkin basis relies on the assertion that every compactly supported divergence-free vector field has a compactly supported smooth stream function. This statement is true (e.g., it follows from H^1_c(R^2)=0), but the one-sentence justification in the text is terse; a short proof or reference would make the approximation property fully rigorous.

Circularity Check

0 steps flagged

No significant circularity: the construction is self-contained and does not reduce to its inputs.

full rationale

The derivation chain for Theorem 1.2 is self-contained. The fixed-dissipation base theorem is not imported from prior work: Section 2 states 'A similar theorem was proved in [27], while we provide an independent proof of the following result' and then proves Theorem 2.1 from Oseen-kernel bounds, a divergence-free cutoff and Galerkin scheme, and a bootstrap iteration. The key uniform-in-epsilon estimate (3.9) follows from the truncation identity (3.5) and the far-field endpoint (3.7), both proved in-paper using the vortex equation and the far-field asymptotics (2.33); no constant is fitted to the target solution. The initial datum enters through the homogeneous far-field boundary condition, which is the natural formulation for self-similar problems, and the result genuinely proves that the constructed flow attains that datum in the t->0 trace. The restriction a>1/3 is explicitly flagged in Remark 1.6 as the threshold where the bootstrap rate (3a-1)/2 in (2.17) becomes positive; this is an analytic limitation, not a disguised assumption of the conclusion. The self-citations [9,10] appear only in the introduction's literature review and are not load-bearing. No prediction reduces by construction to an input, and no fitted parameter is relabeled as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

The proof is a self-contained existence argument. No free parameters are fitted to data, and no new physical entities are introduced. The pseudo-velocity Ũ = U − y/(1+a) is a standard change of variables in self-similar coordinates, not an invented entity. The listed axioms are background harmonic-analysis and PDE ingredients plus the theorem's domain hypotheses.

axioms (8)
  • standard math Fractional heat semigroup kernel bounds (Lemma 2.3): |K_t(z)| ≤ C(t^{1/(1+a)}+|z|)^{-3}, etc.
    Used to control the linear operator L and derive pointwise decay in Theorem 2.1; standard results for the fractional Laplacian, proven in the paper.
  • standard math Lorentz space duality: (L^{p,∞})^* = L^{p',1} for 1<p<∞, and density of C_c^∞ in L^{p',1}.
    Used to pass weak-* limits in Propositions 5.1 and 6.2; stated as Theorem A.1(iv).
  • standard math Córdoba–Córdoba inequality: Λ^s η(f) ≤ η'(f) Λ^s f for convex η.
    Key to the truncation identity (3.5) and the nonnegativity of D_ε(k); cited from [13].
  • standard math Rellich–Kondrachov compactness: W^{1,r}(B_R) compactly embeds into L^2(B_R) for r>1 in 2D.
    Used for strong local convergence of the Galerkin and dissipation sequences.
  • standard math Mean-value property and Liouville-type decay for harmonic functions.
    Used in Lemma 4.1 to identify U_ε = K*Ω_ε by showing the difference is a decaying entire harmonic function.
  • domain assumption u0 ∈ C^1(S^1), divergence-free, and (−a)-homogeneous.
    Fundamental hypothesis of the theorem; the entire construction starts from this angular profile.
  • domain assumption Parameter restriction a > 1/3.
    Needed to make the bootstrap coefficient 3a−1 > 0 in (2.17), yielding smooth profiles with the pointwise far-field decay in Theorem 2.1; the paper flags this in Remark 1.6.
  • standard math Finite-measure embedding of weak-L^p into L^r on bounded sets, and Lorentz–Hölder inequality.
    Used throughout for local compactness, product convergence, and pressure estimates; stated in Theorem A.1.

pith-pipeline@v1.3.0-alltime-deepseek · 22972 in / 30440 out tokens · 267380 ms · 2026-08-01T11:59:02.299414+00:00 · methodology

0 comments
read the original abstract

Let $\frac{1}{3}<a<1$ and let $u_0$ be a $C^1$, divergence-free, $(-a)$-homogeneous vector field on $\mathbb{R}^2\setminus\{0\}$. We construct a forward-in-time self-similar solution of the two-dimensional incompressible Euler equations, \[ u(t,x)=t^{-\frac{a}{1+a}} U\left(\frac{x}{t^{{\frac{1}{1+a}}}}\right), \] with initial datum $u_0$. No smallness or sign assumption is imposed on the initial datum. The construction is a vanishing-dissipation limit of the hypodissipative self-similar profiles. The key is a vorticity profile estimate, uniform in the dissipation parameter, in the critical Lorentz space $L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)$. The resulting Euler solution $u$ has locally finite energy and $\nabla u \in L^\infty_t L^{\frac{2}{1+a},\infty}_x$. Indeed, the velocity is continuous in $L^2_{\text{loc}}$, and the vorticity converges weak-* in $L^{\frac{2}{1+a},\infty}$ at the initial time.

discussion (0)

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