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

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Pith's one-line read For the forced $\alpha$-SQG family, Sobolev non-uniqueness holds exactly below the line $s = \alpha + 2/p$.

desk verdict Forced alpha-SQG non-uniqueness with Sobolev regularity is a real result, and most of the proof is solid, but the alpha=1 branch leans on an omitted fixed-point proof that a referee must fill. read the letter →

arxiv 2502.10274 v1 pith:GKMSJAQ6 submitted 2025-02-14 math.AP

Ángel Castro , Daniel Faraco , Francisco Mengual , Marcos Solera This is my paper · ORCID
classification math.AP MSC 35Q3535Q3135A0276B0335B35
keywords alpha-SQGequationnon-uniquenessunstablevorticesself-similarinstabilitysupercriticalSobolevspacesforced2DEulerglobalsmoothsolutionsVishikmethod
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 adding a fixed force to the generalized surface quasi-geostrophic equations ($\alpha$-SQG, the family interpolating between 2D Euler at $\alpha=0$ and the surface quasi-geostrophic equation at $\alpha=1$) destroys uniqueness in a sharp way: for every $0\le\alpha\le1$ and every Sobolev exponent pair with $s<\alpha+2/p$, there is a force for which uncountably many weak solutions start from zero. The mechanism is a new class of smooth, compactly supported unstable vortices whose linearized dynamics have an exponentially growing mode; in self-similar coordinates this mode becomes a genuine second solution, and careful energy estimates show the nonlinear correction cannot kill it. If the result is correct, the forced version of the conjectured uniqueness threshold is settled at natural sharpness for the whole family, and the same construction reverses in time to give global smooth unforced solutions that are neither rotating nor traveling.

What carries the argument

The central object is a self-similarly nonlinearly unstable vortex: a smooth compactly supported radial vortex $\bar\Theta$ such that, in the self-similar variables $\tau=\frac{1}{ab}\log t$, $X=x/(abt)^{1/a}$, the linearization $L_b$ of the $\alpha$-SQG equation around $\bar\Theta$ has an eigenvalue $\lambda$ with $\Re\lambda>0$ and eigenfunction $W\in U_{jn}$, and the nonlinear correction to $\bar\Theta+\varepsilon\Theta_{\rm lin}+\varepsilon^2\Theta_{\rm cor}$ stays of order $o(e^{\Re\lambda\tau})$. The proof builds $\bar\Theta$ through a piecewise constant ansatz whose instability is decided by a discriminant $\Delta(\sigma)$, a fixed-point regularization, a spectral transfer to self-similar coordinates with a skew-adjoint plus compact decomposition needed for $\alpha=1$, and weighted $Y^m$ energy estimates with an inductive ordering of polar-coordinate derivatives.

What would settle it

Compute the discriminant $\Delta(\sigma)$ of Lemma 4.6 for $\alpha=1$ and $n=2$ by numerical quadrature of the kernels $I_{1,1}$ and $I_{2,1}$ on $\sigma\in(1/2,1)$; Proposition 4.1 predicts $\Delta(1)=\Delta'(1)=0$ and $\Delta''(1)<0$, so $\Delta<0$ for $\sigma$ just below 1, and finding $\Delta\ge0$ on a sequence tending to 1 would refute the claimed unstable vortex, as would an explicit failure of the contraction in Proposition 5.2 for $\alpha=1$.

Watch

Extended reading notes

Core claim

Theorem 1.1 asserts: for every $0\le\alpha\le1$, $s\ge0$, $1\le p\le\infty$ with $s<\alpha+2/p$, there exist $T>0$ and a forcing term $f\in L^1([0,T], W^{s,p}\cap \dot H^{(\alpha-2)/2})$ such that the forced $\alpha$-SQG equation has uncountably many solutions $\theta_\varepsilon\in L^\infty([0,T], W^{s,p}\cap \dot H^{(\alpha-2)/2})$ with $\theta_\varepsilon(0)=0$. The proof exhibits a background vortex $\bar\Theta$, an unstable mode $W$, and a correction $\Theta_{\rm cor}$ so that $\theta_\varepsilon$ is built from $\bar\Theta+\varepsilon\Theta_{\rm lin}+\varepsilon^2\Theta_{\rm cor}$; different values of $\varepsilon$ give different solutions. The same unstable data, run backward in time, produce the corollary of global smooth unforced solutions converging to the vortex, and the regularity is high enough that the non-unique solutions still satisfy the Hamiltonian identity and the renormalization property.

Load-bearing premise

The load-bearing premise is that the SQG regularization fixed point (Proposition 5.2, whose proof is omitted as analogous) actually exists with the stated bounds, together with the $H^m$ time-reversal regularity that depends on the same bootstrap; if that fixed point fails, the smooth unstable vortex used in every theorem disappears.

Editorial extensions

If this is right

  • Non-uniqueness with forcing holds simultaneously in every Sobolev space below the critical line, and by Corollary 1.1 also in the supercritical H\"older scale $\Lambda^{-1}\theta\in C^\gamma$ for $\gamma<1+\alpha$.
  • The non-unique solutions satisfy the Hamiltonian identity and the renormalization property, so neither conservation law singles out a unique flow in the forced supercritical regime.
  • For every $n\ge2$ there exist smooth compactly supported $n$-fold symmetric vortices that are nonlinearly unstable for the unforced equation, with solutions exiting any small $H^m$ neighborhood of the vortex for $m>\alpha+1$.
  • Reversing time on the unstable manifold gives global smooth unforced $\alpha$-SQG solutions that are $n$-fold symmetric, non-stationary, neither rotating nor traveling, and converge to $\bar\Theta$ in $H^m$ as $t\to\infty$.

Reading between the lines

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

  • The discriminant calculation in Proposition 4.1 is carried out for $0\le\alpha<2$, so if the regularization and spectral steps could be extended beyond $\alpha=1$, the non-uniqueness theorem itself would likely extend into the hyperdissipative range $1<\alpha<2$.
  • The physical-time convergence rate of the global solutions of Corollary 1.3 is a power law, since $\tau\sim\log t$ turns $e^{\Re\lambda\tau}$ into $t^{\Re\lambda/(ab)}$; a numerical experiment tracking perturbations backward should measure this exponent, providing a direct test of the instability construction.
  • The inductive polar-coordinate energy scheme is organized so that each derivative estimate uses only lower-index estimates; this suggests the same spaces $Y^m$ could be used to prove instability-driven non-uniqueness for other active scalars whose Cartesian energy estimates fail near the origin.
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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

2 major / 4 minor

Summary. The paper claims non-uniqueness for the forced α-SQG equation (0 ≤ α ≤ 1) in the full supercritical Sobolev regime s < α + 2/p, with forcing in L^1([0,T], W^{s,p} ∩ Ḣ^{(α−2)/2}). The proof follows Vishik's instability strategy: the authors construct a piecewise constant unstable vortex, regularize it to a compactly supported smooth vortex, prove self-similar linear instability via semigroup perturbation, and then prove nonlinear instability via weighted energy estimates. The same machinery yields smooth unstable vortices and, by time reversal, global H^m solutions to the unforced α-SQG equation that are neither rotating nor traveling. The central theorem (Theorem 1.1) and its refinements (Theorems 3.1, 3.4, 3.5) all rest on Theorem 3.3, whose proof in the case α = 1 depends on Proposition 5.2.

Significance. If the full proof can be verified, this is a substantial result: it would establish the sharp Sobolev threshold for non-uniqueness with forcing for the entire α-SQG family, including the SQG case α = 1, and it would provide the first rigorous construction of unstable vortices for 0 < α ≤ 1. The paper is rich in technical content: the kernel asymptotics for I_{n,α} near σ = 1, the discriminant computation for the piecewise constant vortex, the semigroup perturbation argument, and the inductive weighted energy estimates are all presented in considerable detail. The ad hoc definition of the force in (3.6) is a legitimate degree of freedom for the forced non-uniqueness statement, not circular reasoning. However, the α = 1 regularization step contains a load-bearing omitted proof, and one headline corollary is stated more strongly than what is proved.

major comments (2)
  1. [§5.4, Proposition 5.2] Proposition 5.2 is the crucial bridge for the SQG case α = 1, but its proof is omitted with the sentence 'The proof is analogous, and we therefore omit the details.' This is not a cosmetic omission. For α = 1 the expansion in Lemmas 5.3 and 5.6 gives A_0^ε = A_0 + εB_0 and A_1^ε = (log ε)A_1 + B_1, and after the ansatz g = μ + f/log ε the stability equation becomes the coupled system (5.15)–(5.16). The fixed-point map in (5.16) links f, y, and γ through the compatibility condition for the zero-mean operator A_1, and the contraction must control the log ε factors using uniform bounds from Lemmas 5.3 and 5.6. This is exactly where the log divergence of the α = 1 kernel must be tamed, and it is not the same smallness structure as in the α < 1 case of Proposition 5.1, where the error is ε^{1−α}/(1−α). Since Theorems 3.4, 3.5, and ultimately Theorem 1.1 for α = 1 all rely on the smooth vortex from Theorem 3.3, the omitted proof is load-bearing. The authors should supply the full fixed-point argument, either in the main text or in an appendix, including the verification of uniform contraction and the stable inversion of the compatibility condition.
  2. [§1.4, Corollary 1.3 and abstract] The abstract and Corollary 1.3 describe the by-product as 'global smooth solutions,' but the theorem only establishes θ ∈ C([0,∞), H^m ∩ Ḣ^{(α−2)/2}) for a fixed m > α + 1. The authors themselves note in the paragraph after Corollary 1.3 that extension to the global C^∞ case is open. The wording should be corrected to 'global H^m solutions for each fixed m' in the abstract, the introduction, and Corollary 1.3, so that the statement does not overclaim regularity.
minor comments (4)
  1. [§5.1, Lemma 5.2] In the first line of the proof, the expression ε∂_r θ̄_ε = ε(c_1δ_{r_1} + c_1δ_{r_1}) ∗ η_ε should read ε(c_1δ_{r_1} + c_2δ_{r_2}) ∗ η_ε; the second coefficient is a typo.
  2. [§7.5, Proposition 7.2 proof] In the displayed formula for ∂^J( V̄ · ∇Θ), the angular derivative on Θ is written as ∂^{j_1+1}_φ; it should be ∂^{j_2+1}_φ, consistently with ∂^J = ∂^{j_1}_R ∂^{j_2}_φ.
  3. [§1.1, paragraph on upgrades] The sentence 'for every k ∈ N, the other solutions θ_ε can be upgraded to be in C^k_c for positive times' should be qualified as 'for positive times in the interval of construction, with estimates degenerating as t → 0,' since the paper explicitly notes that the estimates deteriorate as t → 0.
  4. [§3.3, Golovkin trick] The presentation of the Golovkin trick is clear, but it would help to state explicitly that the two solutions Θ^+ and Θ^− have the same initial data at τ = −∞ and the same forcing G, since that is the point of the construction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the ad hoc force is a legitimate degree of freedom and the derivation is self-contained apart from an omitted proof that is a correctness risk, not a circular reduction.

full rationale

The derivation chain is not circular. The force is defined by F = -b((a-alpha)+R∂R)barTheta in (3.6), but Theorem 1.1 only asserts existence of some force, so choosing the force from the vortex is an explicitly permitted degree of freedom rather than a prediction forced by the conclusion. The unstable vortex is constructed from a piecewise constant profile via a 2x2 discriminant computation and a fixed-point regularization; the alpha=1 case is treated by a separate log epsilon ansatz and compatibility condition. No parameter is fitted to the target non-uniqueness theorem, and the Sobolev exponents follow from the self-similar scaling in Proposition 2.2 and the choice a < alpha+2/p-s. Self-citations to the authors' [31] supply the baseline alpha=0 argument and analogous fixed-point lemmas, but the alpha>0 branch is derived here rather than imported, and no uniqueness theorem is invoked to force the choice. The flagged weakness is Proposition 5.2, where the SQG fixed-point proof is omitted with the sentence 'The proof is analogous, and we therefore omit the details'; that is a completeness/correctness risk for the alpha=1 branch, not a circular step, because the asserted fixed point is not defined in terms of the theorem's conclusion. The 'global smooth' wording in the abstract also exceeds the proved fixed-H^m statement of Corollary 1.3, but this is an overstatement rather than circularity.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The central claim relies on standard PDE and operator-theoretic background, plus the freedom to choose the forcing term. The construction parameters sigma, c, n, a, b, epsilon, and m are chosen by hand to make the proof work, but none is fitted to empirical data or to the target conclusion. No new physical entities are introduced.

free parameters (7)
  • sigma = chosen in (1/2, 1) sufficiently close to 1
    Radius ratio r1/r2 in the piecewise constant vortex; its choice makes the characteristic polynomial have negative discriminant in Proposition 4.1.
  • c = c = sigma^{-2} - 1
    Imposed by the zero-mean condition (4.15) so the vortex belongs to the space U0.
  • n = integer n >= 2
    Angular frequency of the unstable mode; the discriminant argument requires n >= 2.
  • a = 0 < a < epsilon <= alpha + 2/p - s
    Self-similar time scaling exponent in (2.8); the existence of such a is equivalent to the supercritical condition s < alpha + 2/p.
  • b = 0 < b < Re(lambda)/(m+3)
    Self-similar time parameter chosen small enough to preserve eigenfunction regularity and contraction semigroup estimates.
  • epsilon = small, less than (1/3) min(r1, r2 - r1)
    Regularization scale for the vortex; smallness is needed for the fixed point arguments in Section 5.
  • m = m > alpha + 1, with m >= 5 in the nonlinear section
    Sobolev order in the weighted energy spaces; large enough for well-posedness, embeddings, and the inductive energy estimates.
assumptions (4)
  • standard math Spectral theory of strongly continuous semigroups, compact perturbations, Fredholm operators, and growth bounds (Proposition 6.2 and Appendix A.1).
    Imported background results from Engel-Nagel and Kato, used to pass from linear instability at b = 0 to self-similar instability for small b > 0.
  • standard math Fractional Sobolev embeddings, Rellich-Kondrachov compactness, and commutator estimates for singular integrals.
    Used in Lemma 6.5 for compactness of K when alpha < 1, in Lemma 6.6 for compactness of the SQG commutator, and in Lemmas 7.4 to 7.6 for weighted Sobolev embeddings.
  • domain assumption Local well-posedness for the linearized alpha-SQG system with smooth compactly supported data, as established by Chae-Wu [35].
    Used in Proposition 7.1 and Remark 7.1 to justify existence of the approximate solutions Theta_k^(q).
  • ad hoc to paper The external force is defined ad hoc by F = -b((a-alpha) + R partial_R) Theta_bar, making any vortex a stationary self-similar solution.
    This is a construction choice in Definition 2.1 and (3.6). The theorem only requires existence of some force, not a prescribed force, so this is not circular.

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Pith. "Pith review of Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation." pith.science (2026). https://pith.science/paper/GKMSJAQ6

@misc{pith2026250210274,
  author       = {Pith},
  title        = {Pith review of: Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKMSJAQ6}},
  note         = {Machine review of arXiv:2502.10274}
}
abstract

We prove non-uniqueness of weak solutions to the forced $\alpha$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < \alpha + \frac{2}{p}$, covering the 2D Euler equation ($\alpha = 0$), the Surface Quasi-Geostrophic equation ($\alpha = 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $\alpha$-SQG equation that are neither rotating nor traveling.

Figures

Figures reproduced from arXiv: 2502.10274 by the authors.

Figure 1
Figure 1. Representation of the Sobolev spaces Ws,p in terms of the variables 1/p and s. The green region corresponds to the well-posedness regime s > α + 2/p. The black segment corresponds to the critical line s = α + 2/p. The light red region corresponds to the (expected) non-uniqueness regime s < α + 2/p. The red area s ≤ α + 2/p − ε corresponds to the non-uniqueness with forcing, as established in Theorem 3.1, where ε > 0… view at source ↗

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Cited by 2 Pith papers

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Works this paper leans on

135 extracted references · 59 canonical work pages · cited by 2 Pith papers

  1. [1]

    Albritton, E

    D. Albritton, E. Bru` e, and M. Colombo. Non-uniqueness of Leray solutions of the forced Navier-Stokes equa- tions. Ann. of Math. (2) , 196(1):415–455, 2022

    2022

  2. [2]

    Albritton, E

    D. Albritton, E. Bru` e, M. Colombo, C. De Lellis, V. Giri, M. Janisch, and H. Kwon. Instability and non- uniqueness for the 2D Euler equations, after M. Vishik . Annals of Mathematics Studies. Princeton University Press, Princeton, 2024

    2024

  3. [3]

    D. M. Ambrose, E. Cozzi, D. Erickson, and J. P. Kelliher. Existence of solutions to fluid equations in H¨ older and uniformly local Sobolev spaces. J. Differential Equations , 364:107–151, 2023

    2023

  4. [4]

    W. Ao, J. D´ avila, M. del Pino, M. Musso, and J. Wei. Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation. Trans. Amer. Math. Soc. , 374(9):6665–6689, 2021

    2021

  5. [5]

    V. I. Arnold and B. A. Khesin. Topological methods in hydrodynamics , volume 125 of Applied Mathematical Sciences. Springer, Cham, second edition, [2021] ©2021

    2021

  6. [6]

    Bahouri and J.-Y

    H. Bahouri and J.-Y. Chemin. ´Equations de transport relatives ´ a des champs de vecteurs non-lipschitziens et m´ ecanique des fluides.Arch. Rational Mech. Anal. , 127(2):159–181, 1994

    1994

  7. [7]

    Bauer, P

    M. Bauer, P. Heslin, G. Misio lek, and S. C. Preston. Geometric analysis of the generalized surface quasi- geostrophic equations. Math. Ann., 390(3):4639–4655, 2024

    2024

  8. [8]

    Bedrossian and N

    J. Bedrossian and N. Masmoudi. Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ. Math. Inst. Hautes ´Etudes Sci. , 122:195–300, 2015

    2015

Show all 135 references
  1. [9]

    Beekie, T

    R. Beekie, T. Buckmaster, and V. Vicol. Weak solutions of ideal MHD which do not conserve magnetic helicity. Ann. PDE, 6(1):Paper No. 1, 40, 2020

    2020

  2. [10]

    L. C. Berselli. Vanishing viscosity limit and long-time behavior for 2D quasi-geostrophic equations. Indiana Univ. Math. J. , 51(4):905–930, 2002

    2002

  3. [11]

    Bourgain and D

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

    2015

  4. [12]

    J. P. Bourguignon and H. Brezis. Remarks on the Euler equation. J. Functional Analysis , 15:341–363, 1974

    1974

  5. [13]

    Bressan and R

    A. Bressan and R. Murray. On self-similar solutions to the incompressible Euler equations. J. Differential Equations, 269(6):5142–5203, 2020

    2020

  6. [14]

    Bressan and W

    A. Bressan and W. Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete Contin. Dyn. Syst. , 41(1):113–130, 2021

    2021

  7. [15]

    Bru` e and M

    E. Bru` e and M. Colombo. Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space. Comm. Math. Phys. , 403(2):1171–1192, 2023

    2023

  8. [16]

    Bru` e, M

    E. Bru` e, M. Colombo, and A. Kumar. Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity. arXiv:2408.07934, 2024

    2024 arXiv

  9. [17]

    Bru´ e and Q.-H

    E. Bru´ e and Q.-H. Nguyen. Sobolev estimates for solutions of the transport equation and ODE flows associated to non-Lipschitz drifts. Math. Ann., 380(1-2):855–883, 2021

    2021

  10. [18]

    Buck and S

    M. Buck and S. Modena. Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces. J. Math. Fluid Mech. , 26(2):Paper No. 26, 39, 2024

    2024

  11. [19]

    Buckmaster, C

    T. Buckmaster, C. De Lellis, L. Sz´ ekelyhidi, Jr., and V. Vicol. Onsager’s conjecture for admissible weak solu- tions. Comm. Pure Appl. Math. , 72(2):229–274, 2019

    2019

  12. [20]

    Buckmaster, N

    T. Buckmaster, N. Masmoudi, M. Novack, and V. Vicol. Intermittent convex integration for the 3D Euler equations, volume 217 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, [2023] ©2023

    2023

  13. [21]

    Buckmaster, S

    T. Buckmaster, S. Shkoller, and V. Vicol. Nonuniqueness of weak solutions to the SQG equation. Comm. Pure Appl. Math. , 72(9):1809–1874, 2019

    2019

  14. [22]

    Buckmaster and V

    T. Buckmaster and V. Vicol. Convex integration constructions in hydrodynamics. Bull. Amer. Math. Soc. (N.S.), 58(1):1–44, 2021

    2021

  15. [23]

    Bulut and H

    A. Bulut and H. Dong. Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime. Comm. Math. Phys. , 384(3):1679–1707, 2021

    2021

  16. [24]

    Bulut, M

    A. Bulut, M. K. Huynh, and S. Palasek. Convex integration above the Onsager exponent for the forced Euler equations. arXiv:2301.00804, 2023

    2023 arXiv

  17. [25]

    Bulut, M

    A. Bulut, M. K. Huynh, and S. Palasek. Non-uniqueness up to the Onsager threshold for the forced SQG equation. arXiv:2310.12947, 2023

    2023 arXiv

  18. [26]

    Castro, D

    A. Castro, D. C´ ordoba, and D. Faraco. Mixing solutions for the Muskat problem.Invent. Math., 226(1):251–348, 2021

    2021

  19. [27]

    Castro, D

    A. Castro, D. C´ ordoba, and J. G´ omez-Serrano. Global smooth solutions for the inviscid SQG equation.Mem. Amer. Math. Soc. , 266(1292):v+89, 2020

    2020

  20. [28]

    Castro, D

    A. Castro, D. Faraco, and B. Gebhard. Entropy solutions to macroscopic IPM. arXiv:2309.03637, 2023. To appear in Anal. PDE. INSTABILITY, NON-UNIQUENESS AND GLOBAL SMOOTH SOLUTIONS FOR SQG 69

    2023

  21. [29]

    Castro, D

    A. Castro, D. Faraco, and F. Mengual. Degraded mixing solutions for the Muskat problem. Calc. Var. Partial Differential Equations, 58(2):Paper No. 58, 29, 2019

    2019

  22. [30]

    Castro, D

    A. Castro, D. Faraco, and F. Mengual. Localized mixing zone for Muskat bubbles and turned interfaces. Ann. PDE, 8(1):Paper No. 7, 50, 2022

    2022

  23. [31]

    Castro, D

    A. Castro, D. Faraco, F. Mengual, and M. Solera. A proof of Vishik’s nonuniqueness theorem for the forced 2D Euler equation. arXiv:2404.15995, 2024

    2024 arXiv

  24. [32]

    D. Chae. The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity, 16(2):479–495, 2003

    2003

  25. [33]

    D. Chae, P. Constantin, D. C´ ordoba, F. Gancedo, and J. Wu. Generalized surface quasi-geostrophic equations with singular velocities. Comm. Pure Appl. Math. , 65(8):1037–1066, 2012

    2012

  26. [34]

    D. Chae, P. Constantin, and J. Wu. Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations. Arch. Ration. Mech. Anal. , 202(1):35–62, 2011

    2011

  27. [35]

    Chae and J

    D. Chae and J. Wu. Logarithmically regularized inviscid models in borderline Sobolev spaces. J. Math. Phys. , 53(11):115601, 15, 2012

    2012

  28. [36]

    Cheskidov, M

    A. Cheskidov, M. C. Lopes Filho, H. J. Nussenzveig Lopes, and R. Shvydkoy. Energy conservation in two- dimensional incompressible ideal fluids. Comm. Math. Phys. , 348(1):129–143, 2016

    2016

  29. [37]

    Constantin, T

    P. Constantin, T. D. Drivas, and T. M. Elgindi. Inviscid limit of vorticity distributions in the Yudovich class. Comm. Pure Appl. Math. , 75(1):60–82, 2022

    2022

  30. [38]

    Constantin, W

    P. Constantin, W. E, and E. S. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys. , 165(1):207–209, 1994

    1994

  31. [39]

    Constantin, M

    P. Constantin, M. Ignatova, and H. Q. Nguyen. Inviscid limit for SQG in bounded domains. SIAM J. Math. Anal., 50(6):6196–6207, 2018

    2018

  32. [40]

    Constantin, A

    P. Constantin, A. J. Majda, and E. Tabak. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity, 7(6):1495–1533, 1994

    1994

  33. [41]

    Cordoba, D

    D. Cordoba, D. Faraco, and F. Gancedo. Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal. , 200(3):725–746, 2011

    2011

  34. [42]

    C´ ordoba, M

    D. C´ ordoba, M. A. Fontelos, A. M. Mancho, and J. L. Rodrigo. Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA , 102(17):5949–5952, 2005

    2005

  35. [43]

    C´ ordoba, J

    D. C´ ordoba, J. G´ omez-Serrano, and A. D. Ionescu. Global solutions for the generalized SQG patch equation. Arch. Ration. Mech. Anal. , 233(3):1211–1251, 2019

    2019

  36. [44]

    C´ ordoba, J

    D. C´ ordoba, J. Lucas-Manch´ on, and L. Mart ´ ınez-Zoroa. Strong ill-posedness and non-existence in Sobolev spaces for generalized-SQG. arXiv:2502.06357, 2025

    2025 arXiv

  37. [45]

    C´ ordoba and L

    D. C´ ordoba and L. Mart ´ ınez-Zoroa. Non existence and strong ill-posedness inC k and Sobolev spaces for SQG. Adv. Math., 407:Paper No. 108570, 74, 2022

    2022

  38. [46]

    C´ ordoba and L

    D. C´ ordoba and L. Mart ´ ınez-Zoroa. Non-existence and strong ill-posedness inC k,β for the generalized surface quasi-geostrophic equation. Comm. Math. Phys. , 405(7):Paper No. 170, 53, 2024

    2024

  39. [47]

    C´ ordoba, L

    D. C´ ordoba, L. Mart ´ ınez-Zoroa, and W. S. O˙ za´ nski. Instantaneous continuous loss of regularity for the SQG equation. arXiv:2409.18900, 2024

    2024

  40. [48]

    C´ ordoba, L

    D. C´ ordoba, L. Mart ´ ınez-Zoroa, and W. S. O˙ za´ nski. Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations. Duke Math. J. , 173(10):1931–1971, 2024

    1931

  41. [49]

    C´ ordoba

    D. C´ ordoba. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Annals of Mathe- matics, 148(3):1135–1152, 1998

    1998

  42. [50]

    Crippa and S

    G. Crippa and S. Spirito. Renormalized solutions of the 2D Euler equations. Comm. Math. Phys. , 339(1):191– 198, 2015

    2015

  43. [51]

    M. Dai, V. Giri, and R.-O. Radu. An Onsager type theorem for SQG. arXiv:2407.02582, 2024

    2024 arXiv

  44. [52]

    Dai and Q

    M. Dai and Q. Peng. Non unique weak solutions for forced SQG. arXiv:2310.13537, 2023

    2023 arXiv

  45. [53]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. The Euler equations as a differential inclusion. Ann. of Math. (2) , 170(3):1417–1436, 2009

    2009

  46. [54]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. High dimensionality and h-principle in PDE.Bull. Amer. Math. Soc. (N.S.) , 54(2):247–282, 2017

    2017

  47. [55]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. On turbulence and geometry: from Nash to Onsager. Notices Amer. Math. Soc., 66(5):677–685, 2019

    2019

  48. [56]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. Weak stability and closure in turbulence. Philos. Trans. Roy. Soc. A , 380(2218):Paper No. 20210091, 16, 2022

    2022

  49. [57]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012

    2012

  50. [58]

    R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98(3):511–547, 1989

    1989

  51. [59]

    R. J. DiPerna and A. J. Majda. Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. , 40(3):301–345, 1987. 70 ´ANGEL CASTRO, DANIEL F ARACO, FRANCISCO MENGUAL, AND MARCOS SOLERA

    1987

  52. [60]

    Dolce and G

    M. Dolce and G. Mescolini. Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations. arXiv:2411.18452, 2024

    2024 arXiv

  53. [61]

    P. G. Drazin and W. H. Reid. Hydrodynamic stability. Cambridge University Press, Cambridge, 2004

    2004

  54. [62]

    Duchon and R

    J. Duchon and R. Robert. Inertial energy dissipation for weak solutions of incompressible Euler and Navier- Stokes equations. Nonlinearity, 13(1):249–255, 2000

    2000

  55. [63]

    T. M. Elgindi and I.-J. Jeong. Ill-posedness for the incompressible Euler equations in critical Sobolev spaces. Ann. PDE, 3(1):Paper No. 7, 19, 2017

    2017

  56. [64]

    T. M. Elgindi and N. Masmoudi. L∞ ill-posedness for a class of equations arising in hydrodynamics. Arch. Ration. Mech. Anal., 235(3):1979–2025, 2020

    1979

  57. [65]

    T. M. Elgindi, R. W. Murray, and A. R. Said. Wellposedness and singularity formation beyond the Yudovich class. arXiv:2312.17610, 2023

    2023 arXiv

  58. [66]

    V. Elling. Algebraic spiral solutions of 2d incompressible Euler. J. Differential Equations , 255(11):3749–3787, 2013

    2013

  59. [67]

    Engel and R

    K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations , volume 194 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi...

    2000

  60. [68]

    G. L. Eyink. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D , 78(3-4):222–240, 1994

    1994

  61. [69]

    G. L. Eyink. Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity, 14(4):787–802, 2001

    2001

  62. [70]

    Faraco and S

    D. Faraco and S. Lindberg. Proof of Taylor’s conjecture on magnetic helicity conservation. Comm. Math. Phys. , 373(2):707–738, 2020

    2020

  63. [71]

    Faraco, S

    D. Faraco, S. Lindberg, and L. Sz´ ekelyhidi, Jr. Bounded solutions of ideal MHD with compact support in space-time. Arch. Ration. Mech. Anal. , 239(1):51–93, 2021

    2021

  64. [72]

    Faraco, S

    D. Faraco, S. Lindberg, and L. Sz´ ekelyhidi, Jr. Rigorous results on conserved and dissipated quantities in ideal MHD turbulence. Geophys. Astrophys. Fluid Dyn. , 116(4):237–260, 2022

    2022

  65. [73]

    Faraco, S

    D. Faraco, S. Lindberg, and L. Sz´ ekelyhidi, Jr. Magnetic helicity, weak solutions and relaxation of ideal MHD. Comm. Pure Appl. Math. , 77(4):2387–2412, 2024

    2024

  66. [74]

    F¨ orster and L

    C. F¨ orster and L. Sz´ ekelyhidi, Jr. Piecewise constant subsolutions for the Muskat problem.Comm. Math. Phys., 363(3):1051–1080, 2018

    2018

  67. [75]

    Friedlander and R

    S. Friedlander and R. Shvydkoy. The unstable spectrum of the surface quasi-geostropic equation. J. Math. Fluid Mech., 7:S81–S93, 2005

    2005

  68. [76]

    U. Frisch. Turbulence. Cambridge University Press, Cambridge, 1995. The Legacy of A. N. Kolmogorov

    1995

  69. [77]

    V. Giri, H. Kwon, and M. Novack. The L3-based strong Onsager theorem. arXiv:2305.18509, 2023

    2023 arXiv

  70. [78]

    Giri and R

    V. Giri and R. a.-O. Radu. The Onsager conjecture in 2D: a Newton-Nash iteration. Invent. Math., 238(2):691– 768, 2024

    2024

  71. [79]

    Godard-Cadillac

    L. Godard-Cadillac. Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations. C. R. Math. Acad. Sci. Paris , 359:85–98, 2021

    2021

  72. [80]

    Golovkin

    K. Golovkin. Nonuniqueness of the solutions of certain boundary problems for the equations of hydromechanics. USSR Computational Mathematics and Mathematical Physics , 4(4):212–215, 1964

    1964

  73. [81]

    Gravejat and D

    P. Gravejat and D. Smets. Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation. Int. Math. Res. Not. IMRN , (6):1744–1757, 2019

    2019

  74. [82]

    E. Grenier. On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. , 53(9):1067– 1091, 2000

    2000

  75. [83]

    Hasegawa

    A. Hasegawa. Self-organization processes in continuous media. Adv. in Physics , 34(1):1–42, 1985

    1985

  76. [84]

    He and A

    S. He and A. Kiselev. Small-scale creation for solutions of the SQG equation. Duke Math. J. , 170(5):1027–1041, 2021

    2021

  77. [85]

    I. M. Held, R. T. Pierrehumbert, S. T. Garner, and K. L. Swanson. Surface quasi-geostrophic dynamics. Journal of Fluid Mechanics , 282:1–20, 1995

    1995

  78. [86]

    H¨ older

    E. H¨ older. ¨Uber die unbeschr¨ ankte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Fl¨ ussigkeit.Math. Z. , 37(1):727–738, 1933

    1933

  79. [87]

    E. Hopf. ¨Uber die Anfangswertaufgabe f¨ ur die hydrodynamischen Grundgleichungen.Math. Nachr., 4:213–231, 1951

    1951

  80. [88]

    A. D. Ionescu and H. Jia. Inviscid damping near the Couette flow in a channel.Comm. Math. Phys., 374(3):2015– 2096, 2020

    2015

  81. [89]

    A. D. Ionescu and H. Jia. Non-linear inviscid damping near monotonic shear flows. Acta Math., 230(2):321–399, 2023

    2023

  82. [90]

    A. D. Ionescu and H. Jia. On the nonlinear stability of shear flows and vortices. InICM—International Congress of Mathematicians. Vol. 5. Sections 9–11 , pages 3776–3799. EMS Press, Berlin, [2023] ©2023

    2023

  83. [91]

    P. Isett. A proof of Onsager’s conjecture. Ann. of Math. (2) , 188(3):871–963, 2018. INSTABILITY, NON-UNIQUENESS AND GLOBAL SMOOTH SOLUTIONS FOR SQG 71

    2018

  84. [92]

    Isett and S.-Z

    P. Isett and S.-Z. Looi. A proof of Onsager conjecture for SQG equations. arXiv:2407.02578, 2024

    2024 arXiv

  85. [93]

    Isett and A

    P. Isett and A. Ma. On the conservation laws and the structure of the nonlinearity for SQG and its generaliza- tions. arXiv:2403.08279, 2024

    2024 arXiv

  86. [94]

    Isett and V

    P. Isett and V. Vicol. H¨ older continuous solutions of active scalar equations. Ann. PDE, 1(1):Art. 2, 77, 2015

    2015

  87. [95]

    I.-J. Jeong. Loss of regularity for the 2D Euler equations. J. Math. Fluid Mech. , 23(4):Paper No. 95, 11, 2021

    2021

  88. [96]

    Jeong and J

    I.-J. Jeong and J. Kim. Strong ill-posedness for SQG in critical Sobolev spaces. Anal. PDE , 17(1):133–170, 2024

    2024

  89. [97]

    Jia and V

    H. Jia and V. ˇSver´ ak. Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space? J. Funct. Anal. , 268(12):3734–3766, 2015

    2015

  90. [98]

    T. Kato. Perturbation theory for linear operators . Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition

    1995

  91. [99]

    Kiselev and F

    A. Kiselev and F. Nazarov. A simple energy pump for the surface quasi-geostrophic equation. In Nonlinear partial differential equations, volume 7 of Abel Symp., pages 175–179. Springer, Heidelberg, 2012

    2012

  92. [100]

    Kiselev and V

    A. Kiselev and V. ˇSver´ ak. Small scale creation for solutions of the incompressible two-dimensional Euler equa- tion. Ann. of Math. (2) , 180(3):1205–1220, 2014

    2014

  93. [101]

    O. A. Ladyˇ zenskaja. An example of nonuniqueness in Hopf’s class of weak solutions of the Navier-Stokes equations. Izv. Akad. Nauk SSSR Ser. Mat. , 33:240–247, 1969

    1969

  94. [102]

    Z. Lin. Instability of some ideal plane flows. SIAM Journal on Mathematical Analysis , 35(2):318–356, 2003

    2003

  95. [103]

    M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes. Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence. Arch. Ration. Mech. Anal. , 179(3):353–387, 2006

    2006

  96. [104]

    E. N. Lorenz. The predictability of a flow which possesses many scales of motion. Tellus, 21:289–307, 1969

    1969

  97. [105]

    A. J. Majda and A. L. Bertozzi. Vorticity and incompressible flow , volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002

    2002

  98. [106]

    Marchand

    F. Marchand. Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces Lp or ˙H −1/2. Comm. Math. Phys. , 277(1):45–67, 2008

    2008

  99. [107]

    Masmoudi and W

    N. Masmoudi and W. Zhao. Nonlinear inviscid damping for a class of monotone shear flows in a finite channel. Ann. of Math. (2) , 199(3):1093–1175, 2024

    2024

  100. [108]

    F. Mengual. Non-uniqueness of Admissible Solutions for the 2D Euler Equation with Lp Vortex Data. Comm. Math. Phys. , 405(9):Paper No. 207, 2024

    2024

  101. [109]

    Misio lek and X.-T

    G. Misio lek and X.-T. Vu. On continuity properties of solution maps of the generalized SQG family. Vietnam J. Math. , 52(3):689–698, 2024

    2024

  102. [110]

    Novack and V

    M. Novack and V. Vicol. An intermittent Onsager theorem. Invent. Math. , 233(1):223–323, 2023

    2023

  103. [111]

    L. Onsager. Statistical hydrodynamics. Nuovo Cimento (9) , 6(Supplemento, 2 (Convegno Internazionale di Meccanica Statistica)):279–287, 1949

    1949

  104. [112]

    W. M. F. Orr. The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid. Part II: A Viscous Liquid. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 27:69–138, 1907

    1907

  105. [113]

    F. Otto. Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Math. , 52(7):873–915, 1999

    1999

  106. [114]

    Pedlosky

    J. Pedlosky. Geophysical fluid dynamics . Springer, 1979

    1979

  107. [115]

    Rayleigh

    L. Rayleigh. On the Stability, or Instability, of Certain Fluid Motions. Proceedings of the London Mathematical Society, s1-11:57–72, 1880

  108. [116]

    S. G. Resnick. Dynamical problems in non-linear advective partial differential equations . ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–The University of Chicago

    1995

  109. [117]

    Scheffer

    V. Scheffer. An inviscid flow with compact support in space-time. J. Geom. Anal. , 3(4):343–401, 1993

    1993

  110. [118]

    F. Shao, D. Wei, and Z. Zhang. Self-similar algebraic spiral solution of 2-D incompressible Euler equations. arXiv:2305.05182, 2023

    2023 arXiv

  111. [119]

    Shnirelman

    A. Shnirelman. On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. , 50(12):1261–1286, 1997

    1997

  112. [120]

    Shvydkoy

    R. Shvydkoy. Convex integration for a class of active scalar equations. J. Amer. Math. Soc. , 24(4):1159–1174, 2011

    2011

  113. [121]

    E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , volume 43 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 1993

    1993

  114. [122]

    Sz´ ekelyhidi, Jr

    L. Sz´ ekelyhidi, Jr. Relaxation of the incompressible porous media equation. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 45(3):491–509, 2012

    2012

  115. [123]

    G. I. Taylor. Stability of a viscous liquid contained between two rotating cylinders. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 102(718):289–343, 1923

    1923

  116. [124]

    W. Thomson. 3. Vibrations of a columnar vortex. Proceedings of the Royal Society of Edinburgh , 10:443–456, 1880. 72 ´ANGEL CASTRO, DANIEL F ARACO, FRANCISCO MENGUAL, AND MARCOS SOLERA

  117. [125]

    Tollmien

    W. Tollmien. Ein allgemeines Kriterium der Instabilit¨ at laminarer Geschwindigkeitsverteilungen. Nachr. Ges. Wiss. G¨ ottingen, Math. Phys. Klasse, pages 79–114, 1935

    1935

  118. [126]

    H. Triebel. Theory of Function Spaces. Birkh¨ auser Basel, 1983

    1983

  119. [127]

    M. Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incom- pressible fluid. Part I. arXiv:1805.09426, 2018

    2018 arXiv

  120. [128]

    M. Vishik. Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incom- pressible fluid. Part II. arXiv:1805.09440, 2018

    2018 arXiv

  121. [129]

    Washabaugh

    P. Washabaugh. The SQG equation as a geodesic equation. Arch. Ration. Mech. Anal., 222(3):1269–1284, 2016

    2016

  122. [130]

    Wiedemann

    E. Wiedemann. Weak-strong uniqueness in fluid dynamics. In Partial differential equations in fluid mechanics , volume 452 of London Math. Soc. Lecture Note Ser. , pages 289–326. Cambridge Univ. Press, Cambridge, 2018

    2018

  123. [131]

    Wolibner

    W. Wolibner. Un theor` eme sur l’existence du mouvement plan d’un fluide parfait, homog` ene, incompressible, pendant un temps infiniment long. Math. Z. , 37(1):698–726, 1933

    1933

  124. [132]

    J. Wu. Solutions of the 2D quasi-geostrophic equation in H¨ older spaces. Nonlinear Anal., 62(4):579–594, 2005

    2005

  125. [133]

    Yudovich

    V. Yudovich. Non-stationary flow of an ideal incompressible liquid. USSR Computational Mathematics and Mathematical Physics, 3(6):1407–1456, 1963

    1963

  126. [134]

    X. Zhao. An Onsager-type Theorem for General 2D Active Scalar Equations. arXiv:2412.11094, 2024

    2024 arXiv

  127. [135]

    Zlatoˇ s

    A. Zlatoˇ s. Exponential growth of the vorticity gradient for the Euler equation on the torus.Adv. Math., 268:396– 403, 2015. ´Angel Castro Instituto de Ciencias Matem ´aticas, CSIC-UAM-UC3M-UCM 28049 Madrid, Spain E-mail address: angel castro@icmat.es Daniel Faraco Departamen...

    2015

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