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arxiv: 2605.19716 · v1 · pith:YHPL6CQPnew · submitted 2026-05-19 · 🧮 math.AP

Self-similar blow-up solutions of incompressible Euler equations in mathbb R^d, dgeq 3 with C^(1,1-2/d-) velocity

Feng Shao , Dongyi Wei , Ping Zhang , Zhifei Zhang This is my paper

Pith reviewed 2026-05-20 03:46 UTC · model grok-4.3

classification 🧮 math.AP
keywords self-similar blow-upincompressible Euler equationsaxisymmetric flowsHolder continuityfixed-point methodvorticity transportNewtonian potentialssingularity formation
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The pith

Self-similar blow-up solutions exist for the axisymmetric incompressible Euler equations in dimensions d at least 3, with initial velocity in C^{1,alpha} for any alpha below 1-2/d.

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

The paper constructs self-similar blow-up solutions for the axisymmetric incompressible Euler equations without swirl in R^d where d is at least 3. For any alpha in (0, 1-2/d), the initial velocity belongs to C^{1,alpha} while remaining smooth away from the origin. A sympathetic reader would care because this shows finite-time singularity formation from data that is nearly Lipschitz, a regime close to where global existence remains open. The proof sets up a fixed-point problem on a coupled elliptic-transport system for the self-similar profile, recovering vorticity by transport along characteristics and velocity by Newtonian potentials in an auxiliary (d+4)-dimensional space.

Core claim

For any alpha in (0, alpha_d) with alpha_d = 1-2/d, there exists a self-similar blow-up solution of the axisymmetric incompressible Euler equations without swirl in R^d (d greater than or equal to 3) whose initial velocity satisfies u_0 in C^{1,alpha}(R^d) cap C^infty(R^d minus {0}). The construction relies on a fixed-point framework formulated for the self-similar profile system, which takes the form of a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, whereas the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary (d+4)-dimensional space.

What carries the argument

Fixed-point framework for the self-similar profile system as a coupled elliptic-transport system, with transport along characteristics recovering the vorticity profile and Newtonian potentials in an auxiliary (d+4)-dimensional space reconstructing the velocity.

If this is right

  • Such solutions exist for every dimension d at least 3 and every alpha in (0, 1-2/d).
  • The initial velocity is smooth everywhere except at the origin.
  • The chosen spaces capture the exact singular behavior near the origin and the symmetry axis.
  • The nonlinear operators of transport and elliptic reconstruction leave the spaces invariant.

Where Pith is reading between the lines

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

  • The auxiliary-dimension lifting technique may apply to constructing self-similar profiles in the fully three-dimensional non-axisymmetric Euler equations.
  • Numerical solution of the profile system in the auxiliary space for small d could provide an independent check on the existence of the fixed point.
  • The construction suggests that the threshold for possible singularity formation in Euler equations sits at or below the Holder exponent 1-2/d.

Load-bearing premise

There exist function spaces that remain invariant under the nonlinear composition of the transport operator along characteristics and the elliptic reconstruction via Newtonian potentials in the auxiliary (d+4)-dimensional space while capturing the precise singular behavior near the origin and symmetry axis.

What would settle it

Direct substitution of a candidate profile obtained from the fixed-point iteration back into the original Euler equations shows inconsistency, or the iteration fails to converge in the chosen spaces when alpha approaches 1-2/d from below.

Figures

Figures reproduced from arXiv: 2605.19716 by Dongyi Wei, Feng Shao, Ping Zhang, Zhifei Zhang.

Figure 1
Figure 1. Figure 1: The initial curve and characteristic curves for (2.6) In particular, the implicit constants in (2.16) are independent of µ. Indeed, if 0 < σ ≤ arctan(2d), then η(σ) = 1, and hence χµ(σ) (cos σ) 1+ 1 dµ = (cos σ) −1− 1 dµ ≤ [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
read the original abstract

We investigate the axisymmetric incompressible Euler equations without swirl in $\mathbb R^d$ with $d\geq 3$. For any $\alpha\in(0, \alpha_d)$, where $\alpha_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity fields satisfy $u_0\in C^{1,\alpha}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. Our construction relies on a fixed-point framework formulated for the self-similar profile system, which takes the form of a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, whereas the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge lies in selecting suitable function spaces that remain invariant under such nonlinear compositions, while simultaneously capturing the exact singular behavior near the origin and symmetry axis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper constructs self-similar blow-up solutions to the axisymmetric incompressible Euler equations without swirl in R^d (d ≥ 3). For any α ∈ (0, 1-2/d), there exist solutions whose initial velocity satisfies u_0 ∈ C^{1,α}(R^d) ∩ C^∞(R^d ∖ {0}). The construction proceeds via a fixed-point argument on the self-similar profile system, formulated as a coupled elliptic-transport problem: the transport equation recovers the vorticity profile along characteristics, while the velocity is reconstructed from Newtonian potentials in an auxiliary (d+4)-dimensional space. The key technical step is the identification of function spaces that are invariant under this nonlinear map and that encode the precise singular behavior near the origin and symmetry axis.

Significance. If the fixed-point map is shown to be a contraction on a suitable ball, the result would furnish the first explicit self-similar blow-up examples for the Euler equations in dimensions d ≥ 3 with velocity of Hölder regularity arbitrarily close to the scaling-critical exponent 1-2/d. This would strengthen the known picture of singularity formation by providing a direct construction that respects the axisymmetric symmetry and the precise modulus of continuity, complementing existing numerical and perturbative approaches in lower dimensions.

major comments (3)
  1. [§4.1] §4.1, definition of the space X_α and the ball B_R: the invariance proof under the transport map along characteristics must control the precise Hölder modulus near the axis; the current estimate appears to lose an arbitrary small amount of regularity when integrating the weighted measure induced by the axisymmetric reduction, which would prevent the image from staying inside B_R.
  2. [§5.2] §5.2, Proposition 5.3 (elliptic reconstruction): the Newtonian potential in the auxiliary (d+4)-dimensional space is claimed to map the vorticity profile back into C^{1,α}; however, the singular integral estimates near the origin do not explicitly verify that the resulting velocity gradient satisfies the exact α-Hölder condition without an additional logarithmic loss, which is load-bearing for closing the fixed-point argument.
  3. [§6] §6, contraction mapping: the Lipschitz constant of the fixed-point operator is asserted to be less than 1 for small enough R, but the dependence on α is not quantified; if the constant deteriorates as α → (1-2/d)^-, the existence interval for α may be strictly smaller than claimed.
minor comments (2)
  1. [Introduction] The auxiliary dimension is introduced as (d+4) without a brief geometric motivation in the introduction; adding one sentence explaining the origin of the extra dimensions would improve readability.
  2. [§3] Notation for the weighted measure along the axis is used inconsistently between §3 and §4; a single definition placed early would eliminate confusion.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript concerning self-similar blow-up solutions for the axisymmetric incompressible Euler equations in dimensions d ≥ 3. The points raised concern the technical details of the function spaces, elliptic estimates, and contraction mapping. We address each major comment point by point below, providing clarifications based on the existing arguments in the paper and indicating where we will strengthen the exposition in the revised version.

read point-by-point responses

  1. Referee: [§4.1] §4.1, definition of the space X_α and the ball B_R: the invariance proof under the transport map along characteristics must control the precise Hölder modulus near the axis; the current estimate appears to lose an arbitrary small amount of regularity when integrating the weighted measure induced by the axisymmetric reduction, which would prevent the image from staying inside B_R.

    Authors: We thank the referee for this observation. The space X_α is equipped with weighted Hölder norms whose weights are chosen precisely to compensate for the Jacobian factors and measure distortion arising from the axisymmetric reduction in d dimensions. The transport operator along characteristics preserves the C^{1,α} regularity because the velocity field (which generates the flow) belongs to the ball B_R in X_α, yielding bi-Lipschitz control on the flow map with constants depending only on R. The weighted integration therefore does not produce a loss of Hölder modulus. To eliminate any ambiguity in the current presentation, we will insert a short auxiliary lemma in §4.1 that explicitly tracks the norm under the transport map. revision: partial

  2. Referee: [§5.2] §5.2, Proposition 5.3 (elliptic reconstruction): the Newtonian potential in the auxiliary (d+4)-dimensional space is claimed to map the vorticity profile back into C^{1,α}; however, the singular integral estimates near the origin do not explicitly verify that the resulting velocity gradient satisfies the exact α-Hölder condition without an additional logarithmic loss, which is load-bearing for closing the fixed-point argument.

    Authors: We appreciate the referee drawing attention to the singular-integral details. In the lifted (d+4)-dimensional formulation the Newtonian kernel is more integrable near the origin than in the original dimension, and the estimates in the proof of Proposition 5.3 split the integral into a local part (controlled by the L^1 integrability of the kernel in the auxiliary space) and a far-field part (controlled by the decay of the vorticity profile). This decomposition yields the precise α-Hölder continuity of the velocity gradient without logarithmic loss. We will expand the proof of Proposition 5.3 with the explicit splitting and kernel estimates to make the absence of logarithmic deterioration fully transparent. revision: yes

  3. Referee: [§6] §6, contraction mapping: the Lipschitz constant of the fixed-point operator is asserted to be less than 1 for small enough R, but the dependence on α is not quantified; if the constant deteriorates as α → (1-2/d)^-, the existence interval for α may be strictly smaller than claimed.

    Authors: The Lipschitz constant of the fixed-point operator does depend on α through the constants appearing in the Hölder estimates for the transport and elliptic maps. However, for every fixed α ∈ (0, 1-2/d) these constants remain finite. Consequently, one may always choose the radius R = R(α) sufficiently small so that the contraction constant is strictly less than one. This choice of R does not restrict the admissible range of α; the existence statement holds for every α in the open interval (0, 1-2/d). In the revised manuscript we will add a short remark in §6 that records the α-dependence of the constants and confirms that the smallness of R can always be arranged for each fixed α. revision: partial

Circularity Check

0 steps flagged

Direct fixed-point construction with no reduction to inputs by definition or self-citation

full rationale

The paper constructs self-similar blow-up solutions via a fixed-point argument for the coupled elliptic-transport system in carefully chosen function spaces that encode the C^{1,alpha} regularity and smoothness away from the origin/axis. The abstract and description make clear that the spaces are selected to remain invariant under the nonlinear map (transport along characteristics plus Newtonian potential reconstruction in the auxiliary (d+4)-dimensional space). No equations or steps are shown to reduce the claimed existence to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. This is a standard direct existence proof whose central step is the closure of the function-space ball under the map; it does not rely on renaming known results or importing uniqueness from prior author work as an external theorem. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard elliptic regularity for Newtonian potentials in higher dimensions and on the well-posedness of the transport equation along characteristics; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)
  • standard math Newtonian potentials in (d+4) dimensions yield velocity fields with the required Holder regularity when the vorticity satisfies appropriate decay and symmetry conditions.
    Invoked to reconstruct the velocity profile from the vorticity profile.
  • domain assumption The transport equation along particle trajectories preserves the chosen function space when the velocity is recovered from the elliptic equation.
    Required for the fixed-point map to map the space into itself.

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Works this paper leans on

61 extracted references · 61 canonical work pages · 4 internal anchors

  1. [1]

    J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. , 94 (1984), 61–66

    work page 1984

  2. [2]

    Bedrossian, J

    J. Bedrossian, J. Chen, M. P. Gualdani, S. Ji, V. Vicol and J. Yang, Finite time singularities in the Landau equation with very hard potentials. arXiv:2602.05981, 2026

    work page arXiv 2026

  3. [3]

    Bourgain and D

    J. Bourgain and D. Li, Strong illposedness of the incompressible Euler equation in integer C m spaces. Geom. Funct. Anal. , 25 (2015), 1–86

    work page 2015

  4. [4]

    Bourgain and D

    J. Bourgain and D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces. Invent. Math. , 201 (2015), 97–157

    work page 2015

  5. [5]

    Buckmaster, G

    T. Buckmaster, G. Cao-Labora and J. Gómez-Serrano, Smooth imploding solutions for 3D compressible fluids. Forum Math. Pi , 13 (2025), 753–885

    work page 2025

  6. [6]

    Buckmaster and J

    T. Buckmaster and J. Chen, Blowup for the defocusing septic complex-valued nonlinear wave equation in R4+1. arXiv:2410.15619, 2024

    work page arXiv 2024

  7. [7]

    Cao-Labora, J

    G. Cao-Labora, J. Gómez-Serrano, J. Shi and G. Staffilani, Non-radial implosion for compressible Euler and Navier-Stokes in T3 and R3. Camb. J. Math. , 13 (2025), 753–885

    work page 2025

  8. [8]

    Cao-Labora, J

    G. Cao-Labora, J. Gómez-Serrano, J. Shi and G. Staffilani, Non-radial implosion for the defocusing nonlinear Schrödinger equation in Td and Rd. arXiv:2410.04532, 2024

    work page arXiv 2024

  9. [9]

    Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces

    D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces. Asymptot. Anal., 38 (2004), 339–358

    work page 2004

  10. [10]

    Chen, Remarks on the smoothness of the C 1,α asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations

    J. Chen, Remarks on the smoothness of the C 1,α asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations. Nonlinearity, 37 (2024), Paper No. 065018, 32 pp

    work page 2024

  11. [11]

    Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity I: $C^{\infty}$ 1D Limiting Profiles

    J. Chen, Asymptotically Self-Similar Blowup for 3D Incompressible Euler with C 1,1/3− Velocity I: C ∞ 1D Limiting Profiles. arXiv:2605.15149, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  12. [12]

    Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity II: 3D Profiles, Blowup, and Limiting behavior

    J. Chen, Asymptotically Self-Similar Blowup for 3D Incompressible Euler with C 1,1/3− Velocity II: 3D Profiles, Blowup, and Limiting behavior. arXiv:2605.15130, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  13. [13]

    J. Chen, G. Cialdea, S. Shkoller and V. Vicol, Vorticity blowup in 2D compressible Euler equations. To appear in Duke Math. J. ; arXiv:2407.06455, 2024

    work page arXiv 2024

  14. [14]

    Chen and T

    J. Chen and T. Y. Hou, Finite time blowup of 2D Boussinesq and 3D Euler equations with C 1,α velocity and boundary. Comm. Math. Phys. , 383 (2021), 1559–1667

    work page 2021

  15. [15]

    arXiv:2210.07191

    J. Chen and T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis. arXiv:2210.07191v3, 2023

    work page arXiv 2023

  16. [16]

    Chen and T

    J. Chen and T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data II: Rigorous numerics. Multiscale Model. Simul. , 23 (2025), 25–130

    work page 2025

  17. [17]

    J. Chen, T. Y. Hou and D. Huang, On the finite-time blowup of the De Gregorio model for the 3D Euler equations. Comm. Pure Appl. Math. , 74 (2021), 1282–1350

    work page 2021

  18. [18]

    J. Chen, S. Shkoller and V. Vicol, Smooth and stable Euler implosions. arXiv:2605.00808, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  19. [19]

    Choi, I.-J

    K. Choi, I.-J. Jeong and D. Lim, Global regularity for some axisymmetric Euler flows in Rd. Proc. Amer. Math. Soc. , 154 (2026), 2605–2616

    work page 2026

  20. [20]

    K. Choi, T. Y. Hou, A. Kiselev, G. Luo, V. Šverák and Y. Yao, On the finite-time blowup of a one- dimensional model for the three-dimensional axisymmetric Euler equations. Comm. Pure Appl. Math. , 70 (2017), 2218–2243

    work page 2017

  21. [21]

    Constantin, P

    P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation. Comm. Pure Appl. Math. , 38 (1985), 715–724

    work page 1985

  22. [22]

    C´ ordoba and L

    D. Córdoba and L. Martinez-Zoroa, Blow-up for the incompressible 3D-Euler equations with uniform C 1, 1 2 −ε ∩ L2 force. To appear in Duke Math. J. ; arXiv:2309.08495, 2023

    work page arXiv 2023

  23. [23]

    Córdoba, L

    D. Córdoba, L. Martinez-Zoroa and F. Zheng, Finite time singularities to the 3D incompressible Euler equations for solutions in C ∞(R3 \ {0}) ∩ C 1,α ∩ L2. Ann. PDE , 11 (2025), Paper No. 19, 56 pp. SELF-SIMILAR BLOW-UP SOLUTIONS OF INCOMPRESSIBLE EULER 71

    work page 2025

  24. [24]

    Danchin, Axisymmetric incompressible flows with bounded vorticity

    R. Danchin, Axisymmetric incompressible flows with bounded vorticity. Russian Math. Surveys, 62 (2007), 475–496

    work page 2007

  25. [25]

    De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation

    S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation. J. Statist. Phys., 59 (1990), 1251–1263

    work page 1990

  26. [26]

    T. D. Drivas and T. M. Elgindi, Singularity formation in the incompressible Euler equation in finite and infinite time. EMS Surv. Math. Sci. , 10 (2023), 1–100

    work page 2023

  27. [27]

    Y. Guo, M. Hadžić and J. Jang, Larson-Penston self-similar gravitational collapse. Comm. Math. Phys. , 386 (2021), 1551–1601

    work page 2021

  28. [28]

    Y. Guo, M. Hadžić and J. Jang, Naked singularities in the Einstein-Euler system. Ann. PDE , 9 (2023), Paper No. 4, 182 pp

    work page 2023

  29. [29]

    Y. Guo, M. Hadžić, J. Jang and M. Schrecker, Gravitational collapse for polytropic gaseous stars: self- similar solutions. Arch. Ration. Mech. Anal. , 246 (2022), 957–1066

    work page 2022

  30. [30]

    T. M. Elgindi, Finite-time singularity formation for C 1,α solutions to the incompressible Euler equations on R3. Ann. of Math. (2) , 194 (2021), 647–727

    work page 2021

  31. [31]

    T. M. Elgindi, T.-E. Ghoul and N. Masmoudi, On the stability of self-similar blow-up for C 1,α solutions to the incompressible Euler equations on R3. Camb. J. Math. , 9 (2021), 1035–1075

    work page 2021

  32. [32]

    Elling, Self-similar 2D Euler solutions with mixed-sign vorticity

    V. Elling, Self-similar 2D Euler solutions with mixed-sign vorticity. Commun. Math. Phys. , 348 (2016), 27–68

    work page 2016

  33. [33]

    Gunther, On the motion of fluid in a moving container

    N. Gunther, On the motion of fluid in a moving container. Izvestia Akad. Nauk USSR, Ser. Fiz.–Mat. , 20 (1927), 1323–1348

    work page 1927

  34. [34]

    T. Y. Hou and D. Huang, A potential two-scale traveling wave singularity for 3D incompressible Euler equations. Phys. D , 435 (2022), Paper No. 133257, 27 pp

    work page 2022

  35. [35]

    T. Y. Hou and R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Sci. , 16 (2006), 639–664

    work page 2006

  36. [36]

    T. Y. Hou and S. Zhang, Potential singularity of the axisymmetric Euler equations with C α initial vorticity for a large range of α. Multiscale Model. Simul. , 22 (2024), 1326–1364

    work page 2024

  37. [37]

    Huang, X

    D. Huang, X. Qin, X. Wang and D. Wei, Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda model. Arch. Ration. Mech. Anal. , 248 (2024), Paper No. 22, 65 pp

    work page 2024

  38. [38]

    Huang, X

    D. Huang, X. Qin, X. Wang and D. Wei, Exact self-similar finite-time blowup of the Hou-Luo model with smooth profiles. Comm. Math. Phys. , 406 (2025), aper No. 243, 41 pp

    work page 2025

  39. [39]

    Huang, J

    D. Huang, J. Tong and X. Wang, Self-similar finite-time blowups with singular profiles of the generalized Constantin-Lax-Majda model: theoretical and numerical investigations. arXiv:2603.25104, 2026

    work page arXiv 2026

  40. [40]

    Kato and G

    T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. , 41(1988), 891–907

    work page 1988

  41. [41]

    R. M. Kerr, Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A , 5 (1993), 1725–1746

    work page 1993

  42. [42]

    Lichtenstein, Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze

    L. Lichtenstein, Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze. Math. Z. , 23 (1925), 89–154

    work page 1925

  43. [43]

    Lim and I.-J

    D. Lim and I.-J. Jeong, On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl. Arch. Ration. Mech. Anal. , 249 (2025), Paper No. 32, 31 pp

    work page 2025

  44. [44]

    Luo and T

    G. Luo and T. Y. Hou, Toward the finite-time blowup of the 3D axisymmetric Euler equations: a numerical investigation. Multiscale Model. Simul. , 12 (2014), 1722–1776

    work page 2014

  45. [45]

    Luo and T

    G. Luo and T. Y. Hou, Potentially singular solutions of the 3D axisymmetric Euler equations. Proc. Natl. Acad. Sci., 111 (2014), 12968–12973

    work page 2014

  46. [46]

    A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow . Cambridge Texts Appl. Math., 27. Cambridge Univ. Press, Cambridge, 2002. xii+545 pp

    work page 2002

  47. [47]

    Merle, P

    F. Merle, P. Raphäel, I. Rodnianski, and J. Szeftel, On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles. Ann. of Math. (2) , 196 (2022), 567–778

    work page 2022

  48. [48]

    Merle, P

    F. Merle, P. Raphäel, I. Rodnianski, and J. Szeftel, On the implosion of a compressible fluid II: Singularity formation. Ann. of Math. (2) , 196 (2022), 779–889

    work page 2022

  49. [49]

    Merle, P

    F. Merle, P. Raphäel, I. Rodnianski, and J. Szeftel, On blow up for the energy super critical defocusing nonlinear Schrödinger equations. Invent. Math. , 227 (2022), 247–413

    work page 2022

  50. [50]

    Pak and Y

    H. Pak and Y. Park, Existence of solution for the Euler equations in a critical Besov space B1 ∞,1(Rd). Comm. Part. Differ. Equ. , 29(2004), 1149–1166

    work page 2004

  51. [51]

    F. Shao, D. Wei and Z. Zhang, Self-similar imploding solutions of the relativistic Euler equations. arXiv: 2403.11471, 2024. 72 F. SHAO, D. WEI, P. ZHANG, AND Z. ZHANG

    work page arXiv 2024

  52. [52]

    F. Shao, D. Wei and Z. Zhang, On blow-up for the supercritical defocusing nonlinear wave equation. Forum Math. Pi , 13 (2025), Paper No. e15, 59 pp

    work page 2025

  53. [53]

    F. Shao, S. Wang, D. Wei and Z. Zhang, Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases. arXiv:2501.15701, 2025

    work page arXiv 2025

  54. [54]

    F. Shao, D. Wei and Z. Zhang, Self-similar algebraic spiral solution of 2-D incompressible Euler equations. Ann. PDE , 11 (2025), Paper No. 13, 91 pp

    work page 2025

  55. [55]

    F. Shao, D. Wei and Z. Zhang, Self-similar algebraic spiral vortex sheets of 2-D incompressible Euler equations. Ann. PDE , 12 (2026), Paper No. 10, 76 pp

    work page 2026

  56. [56]

    F. Shao, D. Wei and Z. Zhang, Global regularity of axisymmetric Euler equations without swirl in higher dimensions. Acta Math. Sin. (Engl. Ser.) , 42 (2026), 663–679

    work page 2026

  57. [57]

    Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold

    S. Shkoller, Incompressible Euler Blowup at the C 1, 1 3 Threshold. arXiv:2603.10945, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  58. [58]

    M. R. Ukhovskii and V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. , 32 (1968), 52–61

    work page 1968

  59. [59]

    Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type

    M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. École Norm. Sup. (4) , 32 (1999), 769–812

    work page 1999

  60. [60]

    Y. Wang, M. Bennani, J. Martens, S. Racanière, S. Blackwell, A. Matthews, S. Nikolov, G. Cao-Labora, et al., Discovery of Unstable Singularities. arXiv:2509.14185, 2025

    work page arXiv 2025

  61. [61]

    Wang, C.-Y

    Y. Wang, C.-Y. Lai, J. Gómez-Serrano and T. Buckmaster, Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks. Phys. Rev. Lett. , 130 (2023), Paper No. 244002, 6 pp. (F. Shao) School of Mathematical Sciences, Peking University, Beijing 100871, China Email address : fshao@stu.pku.edu.cn (D. Wei) Sc...

    work page 2023