The role of the dimension in uniqueness results for the stationary quasi-geostrophic system

Diego Chamorro (LaMME); Manuel Fernando Cortez (EPN)

REVIEW 2 major objections 5 minor 23 references

The role of the dimension in uniqueness results for the stationary quasi-geostrophic system

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read In dimensions $n=2,3,4$, the unforced stationary fractional quasi-geostrophic system has no nonzero smooth solutions that satisfy the stated near-critical integrability conditions: the only solution is $\theta\equiv0$.

desk verdict A sound, technically useful Liouville theorem for smooth solutions of the stationary fractional QG system, but the advertised uniqueness for the weak solutions built in Theorem 1 is not established for a substantial part of the parameter range. read the letter →

arxiv 2411.16414 v1 pith:F45NLMBE submitted 2024-11-25 math.AP

Diego Chamorro (LaMME) , Manuel Fernando Cortez (EPN) This is my paper · ORCID

classification math.AP MSC 76D0335A0235B65

keywords quasi-geostrophicequationfractionalLaplacianLiouville-typetheoremuniquenessofweaksolutionsstationaryproblemSobolevspacescriticalintegrabilitydimensiondependence

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's aim is a Liouville-type uniqueness theorem for the stationary fractional quasi-geostrophic equation $(-\Delta)^{\alpha/2}\theta + A[\theta]\cdot\nabla\theta - f=0$ in dimensions $n=2,3,4$, where $A[\theta]$ is a divergence-free singular-integral vector field that is bounded on $L^p$. The theorem says that when the forcing is zero, a smooth solution lying in the natural energy space $\dot H^{\alpha/2}(\mathbb{R}^n)$ and satisfying certain Lebesgue integrability conditions near the Sobolev-critical $L^{2n/(n-\alpha)}$ integrability must be the zero solution. The result matters because the existence proof produces weak solutions by a fixed-point argument that carries no uniqueness information, so any theorem that forces the trivial solution clarifies which steady states are possible. The proof tests the equation against $\theta\phi_R$ and shows that both the commutator remainder and the boundary term vanish as the cut-off radius $R\to\infty$, forcing the dissipation term to vanish. The statement is conditional on smoothness, and the paper explicitly notes that regularity of the weak solutions is open for $0<\alpha\le1$ and needs an extra $L^\infty$ assumption for $1<\alpha\le(n+2)/3$.

What carries the argument

The key object is the truncated energy identity obtained by testing (1.1) against $\theta\phi_R$, where $\phi_R$ is a smooth cut-off equal to $1$ on $|x|\le R/2$ and supported in $|x|\le R$. It takes the form (3.5): $$\int_{B_{R/2}} |(-\$\Delta$)^{\$\alpha$/4}\$\theta$|^2\,dx \le I_1 + I_2,$$ with $$I_1=\int_{\mathbb{R}^n} (-\$\Delta$)^{\$\alpha$/4}\$\theta$\left[\big((-\$\Delta$)^{\$\alpha$/4}\$\theta$\big)\phi_R - (-\$\Delta$)^{\$\alpha$/4}(\$\theta$\phi_R)\right]dx$$ and $$I_2=\frac12\int_{\mathbb{R}^n} (A[\$\theta$]\$theta^{2}$)\cdot\nabla\phi_R\,dx.$$ The task is to show $I_1,I_2\to0$ as $R\to\infty$. $I_1$ is a commutator remainder controlled by the fractional Leibniz rule (Kato-Ponce) plus complex interpolation, which is where the near-critical space $L^{(2n-\epsilon)/(n-\alpha)}$ enters; $I_2$ is a boundary term controlled by H\"older inequalities whose admissible exponents change with $n$ and $\alpha$. The role of the dimension is encoded in those exponent ranges: low-$\alpha$ requires an improved integrability with parameter $\nu$, while high-$\alpha$ in dimensions $2,3$ requires $L^{3n/(n-1)}$.

What would settle it

Exhibit a single nonzero smooth function $\theta\in\dot H^{\alpha/2}(\mathbb{R}^n)$ with $n\in\{2,3,4\}$ that satisfies $(-\Delta)^{\alpha/2}\theta+A[\theta]\cdot\nabla\theta=0$ and the stated Lebesgue integrability conditions; any such function would falsify the uniqueness theorem. A concrete first calculation would be to test the radial reduction in $n=2$, $\alpha=1$, where the theorem predicts that no smooth $L^{4-\epsilon}$ solution exists and the cut-off identity becomes a one-dimensional integral check.

Watch

Extended reading notes

Core claim

The central claim is a dimension-dependent Liouville theorem. Take $n=2,3,4$, $f=0$, and a smooth solution $\theta\in\dot H^{\alpha/2}(\mathbb{R}^n)$ of $(-\Delta)^{\alpha/2}\theta + A[\theta]\cdot\nabla\theta=0$. Assume $\theta\in L^{(2n-\epsilon)/(n-\alpha)}$ for a small $\epsilon>0$. The theorem adds: if $0<\alpha<n/3$, also $\theta\in L^{(2n+\nu)/(n-\alpha)}$ with $(n-3\alpha)<\nu<1+(n-3\alpha)$; if $(n+2)/3\le\alpha<2$ and $n=2,3$, also $\theta\in L^{3n/(n-1)}$; in the intermediate regime $n/3\le\alpha<(n+2)/3$ (and for $n=4$ with $\alpha\ge4/3$), no further hypothesis is needed. Under these conditions the only solution is $\theta\equiv0$, meaning a nontrivial steady state must be driven by the forcing.

Load-bearing premise

The load-bearing premise is that the solution is smooth; the paper itself leaves regularity open for $0<\alpha\le1$ and requires an extra $L^\infty$ bound for $1<\alpha\le(n+2)/3$, so for a large part of the parameter range the theorem does not apply to the weak solutions whose existence was proved.

Editorial extensions

If this is right

For $n=2,3,4$ with $f=0$, the only smooth solution in $\dot H^{\alpha/2}$ satisfying the stated integrability is zero, so nonzero steady states in this class must be sustained by the forcing.
For $n=2,3$ and $\alpha\ge(n+2)/3$, the extra condition reduces to $L^{3n/(n-1)}$, which for $n=3$ is $L^{9/2}$, the critical space appearing in known Liouville theorems for stationary Navier-Stokes.
For the intermediate range $n/3\le\alpha<(n+2)/3$ (and for $n=4$, $\alpha\ge4/3$), the Sobolev-critical information alone is enough and no additional Lebesgue assumption is required.
The proof of the $I_1$ estimate uses $\epsilon>0$; the paper leaves open whether the exact critical space $L^{2n/(n-\alpha)}$ would suffice.
The paper notes the same techniques extend to dimensions $n\ge5$, so the restriction to $n=2,3,4$ is not a limitation of the method.

Reading between the lines

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

Because the proof uses only the divergence-free condition and the $L^p$-boundedness of $A[\theta]$, the same cut-off identity should yield analogous Liouville theorems for other stationary fractional transport systems with a divergence-free velocity field.
If the missing regularity for $0<\alpha\le1$ is later established, the uniqueness result would automatically extend from smooth solutions to the weak solutions produced by the existence theorem.
The sharpness of the $\epsilon$ and $\nu$ conditions could be tested by searching for nonzero steady states at exactly the critical exponent $L^{2n/(n-\alpha)}$; finding one would show the extra integrability is essential.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main result is real: a dimension-dependent Liouville theorem for the stationary fractional quasi-geostrophic equation in n=2,3,4, under extra Lebesgue integrability. The cutoff-and-interpolation machinery is standard, but the adaptation to the fractional stationary system is careful, and as far as I can check, the estimates in Section 3 are coherent for smooth solutions. The paper also gives an honest existence proof via the Leray-Schauder/Schaefer scheme, and the a priori energy estimate is clean. This is a useful technical contribution to the Liouville literature for active scalar equations, and it does not overclaim the resolution of the critical open problem with only H-dot^{alpha/2} integrability.

The soft spot is real and is exactly where the reader's stress-test lands. Theorem 2 assumes theta is smooth, and the paper itself states that regularity for 0(n+2)/3; for 1

Proportionally, these are scope gaps, not signs of a flawed core. The central estimates are defensible and the authors are transparent about the regularity limitations. This paper deserves a serious referee: the result is meaningful, the method is sound for the smooth case, and the required revisions are editorial and mathematical tightening, not a restart. I would recommend sending it to peer review and asking the authors to align the abstract with the theorem's actual hypotheses, fix Theorem 1's range, and clean up the small errors.

Referee Report

2 major / 5 minor

Summary. The paper studies the stationary fractional quasi-geostrophic system (-Δ)^{α/2}θ + A[θ]·∇θ - f = 0 on R^n for 0<α<2. Theorem 1 claims existence of weak solutions in Ḣ^{α/2}(R^n) for f∈Ḣ^{-α/2}(R^n). Theorem 2, the main result, asserts that for n=2,3,4 and f=0, any smooth solution satisfying additional Lebesgue integrability conditions (close to the critical Sobolev exponent, with dimension-dependent extras for small and large α) must be identically zero. The proof uses a cutoff function, the fractional Leibniz rule, interpolation, and Hölder estimates to show that the localized energy ∫_{B_{R/2}} |(-Δ)^{α/4}θ|^2 vanishes as R→∞. Appendix A constructs weak solutions via a regularized Leray-Schauder argument, and Appendix B develops partial regularity results. The manuscript explicitly acknowledges that regularity for 0<α≤1 is open.

Significance. If the scope gap identified below is resolved, the paper would provide a clean Liouville-type theorem for a fractional transport equation, with explicit dimension-dependent exponents that are tracked carefully through the proof. The argument is self-contained and uses no fitted constants or circular reasoning; the authors are transparent about the smoothness assumption and the open regularity problem. The existence proof is standard, and the dimension-dependent thresholds (n/3 and (n+2)/3) are natural. The main advertised claim, however, is currently stronger than what is proved: the abstract promises uniqueness of weak solutions, while the theorem only covers solutions already assumed smooth in part of the parameter range.

major comments (2)

[Abstract; Section 1 (after Theorem 2); Appendix B] The abstract and introduction advertise uniqueness of weak solutions, but Theorem 2 assumes that θ is smooth. The authors state explicitly after Theorem 2 that for 0<α≤1 the regularity of the weak solutions obtained in Theorem 1 is 'a completely open problem', and Appendix B establishes smoothness only for α>(n+2)/3, with an additional L∞ assumption on θ and on A in the case 1<α≤(n+2)/3. Consequently, for 0<α≤1 (and for 1<α≤(n+2)/3 without that extra assumption) the uniqueness conclusion is not proved for the weak solutions whose existence is claimed; it holds only for solutions that are already assumed smooth. This is a genuine scope gap between the advertised result and the proved theorem; please narrow the abstract and introduction accordingly, or supply a regularity theorem covering the missing range.
[Appendix A, Theorem 3 and Theorem 1] The existence proof invokes Schaefer's fixed-point theorem (Theorem 3) which is stated only for 1<α<2, while Theorem 1 claims existence for all 0<α<2. The auxiliary estimates in Propositions A.1-A.3 appear adaptable to α≤1 by appropriate choices of the parameters σ and s, but as written the range 0<α≤1 is not covered by the stated fixed-point theorem. Please extend the statement of Theorem 3 (or modify the proof of Theorem 1) so that the full claimed range is justified.

minor comments (5)

[Section 3.1, paragraph after Eq. (3.6)] The sentence 'α − n < 0 since 1 < α < 2 and n ≥ 2' is incorrect in the small-α case covered by Theorem 2; the conclusion α−n<0 follows from 0<α<2 and n≥2, and the text should say so.
[Section 3.1, Eq. (3.6) and subsequent Hölder application] The Hölder exponent (4n−2ε)/(2α−ε) requires 2α−ε>0, so the proof should state that ε is chosen with 0<ε<2α (and small enough for the exponent to exceed 1); without this, the estimate for |I1| is not valid for α very small.
[Appendix B, estimate (B.1)] The inequality ||A[θ]||_{Ḣ^{α/2}} ≤ C||θ||_{Ḣ^{α/2}} is used without proof; this does not follow directly from (1.2) unless one assumes that A commutes with fractional derivatives (as for the Riesz transforms in the actual SQG model) or includes Sobolev boundedness as an additional hypothesis.
[Theorem 2 and Abstract] The phrase 'unique solution' / 'uniqueness of weak solutions' is potentially misleading, since the result is a Liouville theorem stating that zero is the only solution in the class; 'only solution' would be clearer.
[Throughout displayed equations] There are LaTeX artifacts such as '/bracehtipupleft' and '/bracehtipdownright' in Eq. (3.5) and in Appendix A; these should be removed in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the uniqueness conclusion is derived from the equation and auxiliary integrability assumptions via standard inequalities; the acknowledged regularity gaps are scope issues, not circular reasoning.

full rationale

The paper's derivation chain is self-contained. Theorem 2 is proved by multiplying the stationary equation (1.1) with f=0 by θϕ_R, integrating by parts, and showing that the two error terms I1 and I2 tend to zero as R→∞ using the assumed Lebesgue integrability, Sobolev embeddings, the fractional Leibniz/Kato-Ponce rule, complex interpolation, and the boundedness and divergence-free properties of A[θ]. No parameter is fitted to data, no quantity in the conclusion θ≡0 is imported as an assumption, and no load-bearing step relies on a self-citation: the cited works are standard external tools (Grafakos–Oh, Naibo–Thomson, Bergh–Löfström, Brezis, Lemarié-Rieusset) or unrelated Liouville results for Navier–Stokes used only for comparison. The integrability hypotheses such as θ ∈ L^{(2n-ε)/(n-α)} and θ ∈ L^{(2n+ν)/(n-α)} are genuinely additional global-integrablity/decay information and are not equivalent, by construction, to the conclusion that θ is identically zero. The manuscript itself explicitly flags the real limitations: Theorem 2 assumes θ is smooth, and the text states that for small α (in particular 0<α≤1) the regularity of the weak solutions obtained via Theorem 1 is a completely open problem, while Appendix B obtains smoothness only for α>(n+2)/3 and, in the range 1<α≤(n+2)/3, only under an additional L∞ assumption. In addition, the Schaefer fixed-point theorem is stated for 1<α<2, leaving a formal gap in Theorem 1's stated coverage of 0<α≤1. These are scope and correctness gaps concerning which solutions the theorem actually covers; they do not make the derivation circular. The conditional statement 'smooth solutions in the stated L^p spaces must vanish' is derived from the equation and is not assumed as its own conclusion. Therefore no circular step is present and the score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the smoothness assumption on theta, the technical integrability exponents epsilon and nu, and standard theorems from harmonic analysis and fixed-point theory. There are no fitted empirical parameters and no new physical entities. The assumptions on A are the standard quasi-geostrophic modeling framework, not inventions of this paper.

free parameters (2)

epsilon (epsilon parameter in integrability exponent) = 0 < epsilon << 1 (arbitrary small)
Introduced in Theorem 2 to make the cutoff term I1 decay: the exponent (alpha-n)epsilon/(4n-2epsilon) is negative only when epsilon>0, and setting epsilon=0 gives zero decay. This is a hand-picked technical parameter, not fitted to data.
nu (integrability exponent parameter) = nu in (n-3alpha, 1+n-3alpha)
Additional Lebesgue exponent used in the small-alpha case so that the Hoelder exponent 3alpha+nu-n is positive and the R-power in the I2 estimate is negative. It is a tuning parameter chosen to make the estimates work, not an empirical quantity.

assumptions (5)

ad hoc to paper The solution theta in Theorem 2 is assumed smooth
Smoothness is not guaranteed by Theorem 1; Appendix B only proves regularity for alpha>(n+2)/3, or for 1<alpha with an additional L^infinity assumption, and leaves 0<alpha<=1 open. This is the main extra regularity assumption of the uniqueness proof.
domain assumption A[theta] is a divergence-free singular integral operator bounded on L^p for all 1<p<infinity (assumptions (1.2) and (1.3))
This models the quasi-geostrophic drift as a Riesz-type velocity and is used in the energy identity, in the I2 estimates, and in the fixed-point construction. It is stated as part of the system, not proved.
standard math Complex interpolation theorem (Bergh-Lofstrom, Theorem 6.4.5) yields the space identity used in (3.7)
The interpolation between Hdot^{alpha/2} and L^{(2n-epsilon)/(n-alpha)} is invoked to control the norm of (-Delta)^{alpha2/4}theta in L^{p2}. This is a standard external result.
standard math Kato-Ponce fractional Leibniz rule (Lemma 2.2)
Used at (3.6) to localize the fractional Laplacian acting on the product theta phi_R. Cited from the literature and not proved in the paper.
standard math Schaefer/Leray-Schauder fixed point theorem and Rellich-Kondrashov compactness
Used in Appendix A to construct weak solutions of the regularized problem and pass to the limit. These are standard functional analysis tools.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The role of the dimension in uniqueness results for the stationary quasi-geostrophic system." pith.science (2026). https://pith.science/paper/F45NLMBE

@misc{pith2026241116414,
  author       = {Pith},
  title        = {Pith review of: The role of the dimension in uniqueness results for the stationary quasi-geostrophic system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F45NLMBE}},
  note         = {Machine review of arXiv:2411.16414}
}

read the original abstract

In this paper, we study a Liouville-type theorem for the stationary fractional quasi-geostrophic equation in various dimensions. Indeed, our analysis focuses on dimensions n = 2, 3, 4 and we explore the uniqueness of weak solutions for this fractional system. We demonstrate here that, under some specific Lebesgue integrability information, the only admissible solution to the stationary fractional quasi-geostrophic system is the trivial one and this result provides a comprehensive understanding of how the dimension in connection to the fractional power of the Laplacian influences the uniqueness properties of weak solutions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Bahrami and H

F. Bahrami and H. F azli. On the steady solutions of the inviscid two-dimensional qua si- geostrophic equation . Journal of Mathematical Analysis and Applications Volume 515, Issue 2, 126425, (2022). 19

work page 2022
[2]

Bergh and J

J. Bergh and J. L¨ofstr¨om. Interpolation spaces. An introduction. , Springer–Verlag, (1976)

work page 1976
[3]

H. Brezis . Analyse Fonctionnelle, Th´ eorie et applications. Dunod, (1999)

work page 1999
[4]

Chae and T

D. Chae and T. Yoneda. On the Liouville theorem for the stationary Navier-Stokes eq uations in a critical space . J. Math. Anal. Appl. 405: 706-710, (2013)

work page 2013
[5]

Chae and J

D. Chae and J. Wolf. On Liouville type theorems for the steady Navier- Stokes equa tions in Rn. Journal of Diﬀerential Equations, Volume 261, Issue 10, Page s 5541-5560, (2016)

work page 2016
[6]

Constantin

P. Constantin. Energy spectrum of quasigeostrophic turbulence , Phys. Rev. Lett. 89, 184501, (2002)

work page 2002
[7]

Constantin, A

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

work page 1994
[8]

C´ordoba, D

A. C´ordoba, D. C´ordoba and F. Gancedo. Uniqueness for SQG patch solutions , Trans. Amer. Math. Soc. Ser. B, 5, 1–31, (2018)

work page 2018

Show all 23 references

[9]

H. Dong. Dissipative quasi-geostrophic equations in critical Sobo lev spaces: smoothing eﬀect and global well-posedness. Discrete Contin. Dyn. Syst. 26, 1197–1211, (2010)

2010
[10]

G.P. Galdi. An introduction to the mathematical theory of the Navier-St okes equations. Steady- state problems. Second edition. Springer Monographs in Mathematics. Sprin ger, New York (2011)

2011
[11]

Grafakos and S

L. Grafakos and S. Oh, The Kato-Ponce inequality , Comm. Partial Diﬀerential Equations 39, no. 6, 1128–1157, (2014)

2014
[12]

O. Jarr´ın. A remark on the Liouville problem for stationary Navier-Stok es equations in Lorentz and Morrey spaces . J. Math. Anal. Appl, Volume 486, Issue 1 (2020)

2020
[13]

G. Koch, N. Nadirashvili, G. Seregin & V. Sverak. Liouville theorems for the Navier-Stokes equations and applications . Acta Mathematica, 203: 83- 105 (2009)

2009
[14]

Kozono, Y

H. Kozono, Y. Terasawab & Y. W akasugib. A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensi ons. Journal of Functional Analysis, 272: 804-818 (2017)

2017
[15]

Lemari´ e-Rieusset

P.G. Lemari´ e-Rieusset. The Navier-Stokes Problem in the 21st Century . Chapman & Hall/CRC, (2016)

2016
[16]

Michael, A

S. Michael, A. Jolly, A. Kumar and V.R. Martinez. On the Existence, Uniqueness, and Smoothing of Solutions to the Generalized SQG Equations in Cr itical Sobolev Spaces . Commun. Math. Phys. volume 387, pages 551–596, (2021)

2021
[17]

Naibo and A

V. Naibo and A. Thomson. Coifman-Meyer multipliers: Leibniz-type rules and applica tions to scattering of solutions to PDEs . Transactions of the AMS, Volume 372, Number 8, p. 5453–5481 , (2019)

2019
[18]

Pedlosky

J. Pedlosky. Geophysical Fluid Dynamics , Springer, New York, (1987)

1987
[19]

S. Resnick. Dynamical problems in nonlinear advective partial diﬀeren tial equations, Ph.D. The- sis, University of Chicago, (1995)

1995
[20]

G. Seregin. Liouville type theorem for stationary Navier-Stokes equati ons. Nonlinearity, 29 : 2191 :2195 (2015)

2015
[21]

Smith et al

K.S. Smith et al. . Turbulent diﬀusion in the geostrophic inverse cascade . J. Fluid Mech. 469, 13-48, (2002)

2002
[22]

N. Su. The fractional Boussinesq equation of groundwater ﬂow and it s applications . Journal of Hydrology Volume 547, Pages 403-412, (2017)

2017
[23]

W angand J

Y. W angand J. Xiao. A Liouville problem for the stationary fractional Navier-St okes-Poisson system. J. Math. Fluid Mech. 20: 485-498 (2018). 20

2018

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bahrami and H

F. Bahrami and H. F azli. On the steady solutions of the inviscid two-dimensional qua si- geostrophic equation . Journal of Mathematical Analysis and Applications Volume 515, Issue 2, 126425, (2022). 19

work page 2022

[2] [2]

Bergh and J

J. Bergh and J. L¨ofstr¨om. Interpolation spaces. An introduction. , Springer–Verlag, (1976)

work page 1976

[3] [3]

H. Brezis . Analyse Fonctionnelle, Th´ eorie et applications. Dunod, (1999)

work page 1999

[4] [4]

Chae and T

D. Chae and T. Yoneda. On the Liouville theorem for the stationary Navier-Stokes eq uations in a critical space . J. Math. Anal. Appl. 405: 706-710, (2013)

work page 2013

[5] [5]

Chae and J

D. Chae and J. Wolf. On Liouville type theorems for the steady Navier- Stokes equa tions in Rn. Journal of Diﬀerential Equations, Volume 261, Issue 10, Page s 5541-5560, (2016)

work page 2016

[6] [6]

Constantin

P. Constantin. Energy spectrum of quasigeostrophic turbulence , Phys. Rev. Lett. 89, 184501, (2002)

work page 2002

[7] [7]

Constantin, A

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

work page 1994

[8] [8]

C´ordoba, D

A. C´ordoba, D. C´ordoba and F. Gancedo. Uniqueness for SQG patch solutions , Trans. Amer. Math. Soc. Ser. B, 5, 1–31, (2018)

work page 2018