The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

Joan Mateu; Joan Orobitg; Marc Maga\~na

REVIEW 2 major objections 3 minor 1 cited by

The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

T0 review · 2 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Vortex patches in the quasi-geostrophic shallow-water equations keep C^{1,γ} boundaries for all time, and the equations converge to 2D Euler flow as the Rossby radius grows.

desk verdict The convergence-to-Euler theorem is the real, well-supported contribution; the global vortex-patch theorem is partially delegated to the Euler literature and needs a referee to ask for the missing Gronwall details. read the letter →

arxiv 2602.22767 v3 pith:HCVXUKHZ submitted 2026-02-26 math.AP

Marc Magaña , Joan Mateu , Joan Orobitg This is my paper · ORCID

classification math.AP MSC 35Q3576B4735B65

keywords quasi-geostrophicshallow-waterequationsvortexpatchesboundaryregularityHölderspacescontourdynamicsmodifiedBesselfunctionsRossbyradiusconvergencetoEuler

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

The paper proves that the vortex-patch boundary-regularity theorem, known for the 2D Euler equations, carries over to the quasi-geostrophic shallow-water (QGSW) equations. It shows that if the initial patch boundary is C^{1,γ}, the unique weak solution remains the characteristic function of a domain with C^{1,γ} boundary for all time. It also proves that, as the inverse Rossby radius ε tends to zero, QGSW solutions converge to the corresponding Euler solution in little Hölder spaces, uniformly on a time interval. This matters because QGSW is a physically relevant one-parameter family that reduces to Euler when ε=0, yet its kernel is non-homogeneous and does not fit previous Euler-type vortex-patch frameworks.

What carries the argument

The velocity kernel K(x)= (ε/2π)(x^⊥/|x|) K_1(ε|x|), with K_1 a modified Bessel function, is non-homogeneous. The central technical device is the decomposition ∇K = S^{(1)} + S^{(2)}: the singular part S^{(1)} has components of order 1/|x|^2 with zero spherical mean, exactly as the Euler kernel, while S^{(2)} is integrable and contributes only additive constants to the estimates. This lets the classical Euler vortex-patch proof (via the contour dynamics equation for the boundary curve and a defining-function Gronwall argument) be reused, and it yields the uniform-in-ε estimates that prove convergence to Euler.

What would settle it

Compute an explicit bound for ||∇v||_∞ for a QGSW vortex patch in terms of the defining function's Hölder seminorm and check whether the S^{(2)} contribution remains an additive constant. If that contribution grows non-additively with |∇φ|_γ, then the Gronwall inequality would produce a super-double-exponential bound and the global-regularity conclusion would fail; a numerical test with a patch whose boundary has large curvature and small |∇φ|_inf would reveal finite-time corner formation.

Watch

Extended reading notes

Core claim

For every bounded initial domain with boundary of class C^{1,γ} (0<γ<1), there is a unique weak solution to QGSW of the form q(x,t)=χ_{Ω_t}(x), where Ω_t is a bounded domain whose boundary remains in C^{1,γ} for all time. Independently, for initial data in the little Hölder space c^γ_c, the unique QGSW solution converges to the unique 2D Euler solution in the C^γ norm as ε→0, uniformly for t in a time interval that does not shrink to zero.

Load-bearing premise

The global-in-time statement of Theorem 1.1 depends on the assertion that the non-homogeneous, integrable part of the kernel enters the Gronwall estimate only additively and therefore does not change the double-exponential bound that prevents finite-time blow-up of the boundary Hölder norm; the proof of this is deferred to the Euler case.

Editorial extensions

If this is right

If the theorems are correct, the quasi-geostrophic shallow-water model has the same vortex-patch boundary persistence as 2D Euler: initially smooth patch boundaries stay smooth forever, so analytical and numerical studies of QGSW patch dynamics can rely on that regularity.
The convergence result rigorously justifies the formal limit of QGSW to 2D Euler as the Rossby deformation radius becomes infinite (ε→0), at least for initial data in little Hölder spaces and for a common time interval.
The kernel decomposition suggests a general principle: a non-homogeneous kernel that differs from an odd homogeneous kernel by an L^1 additive correction inherits the Euler patch-regularity theorem, provided the correction satisfies suitable gradient bounds.
The uniform-in-ε a priori estimates imply that the QGSW solutions exist and remain bounded in C^γ on a time interval independent of ε, which is the key quantitative input for the convergence proof.

Reading between the lines

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

The proof's reliance on the Euler global-regularity mechanism indicates that the same double-exponential Gronwall strategy would work for any kernel of the form 'odd homogeneous kernel + integrable perturbation', yielding a broader class of active scalar equations with patch persistence; this is a testable extension beyond the specific Bessel kernel.
The convergence result is stated in little Hölder spaces, not for vortex patches; an open question directly suggested by the paper is whether the vortex-patch boundaries themselves converge in C^{1,γ} as ε→0, with a rate controlled by the same kernel difference estimates.
The a priori bound of Proposition 5.4 is logarithmic in the boundary Hölder seminorm; if the S^{(2)} part were to contribute superlinearly in that seminorm to the Gronwall estimate, the bound would not close, so the paper implicitly asserts a precise cancellation whose verification is the main technical risk.
The proof of Theorem 6.1 uses a dense subclass of smooth functions to interchange limits; an inference is that the convergence in C^γ may hold for all C^γ data, but the little-Hölder condition is essential for the proof, suggesting that convergence may fail in the full Hölder space without that vanishing-oscillation assumption.

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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

Bottom line: Theorem 6.1, the little-Hölder convergence to 2D Euler as ε→0, is the genuinely new part of this paper and it looks solid. The vortex-patch persistence theorem, Theorem 1.1, is largely parallel to Tan–Xue–Xue [22], as the authors honestly acknowledge, and its global-in-time half is not fully proved here. The paper still deserves serious refereeing, but the referee should insist on seeing the missing arguments.

What is actually new: the convergence theorem on a uniform time interval, and the ε-independent kernel estimates that support it. Lemma 6.3 is proved in real detail, with careful splitting of the kernel difference into the near and far regions, and the constants are checked to be independent of ε. Proposition 5.5, the commutator estimate, is also proved in full and appears correct. The decomposition of ∇K into a singular zero-mean part S^(1) and an L^1 part S^(2) is genuinely useful, and the authors verify the ε-independence of the constants rather than just citing the Euler case.

Credit is due for the candor: the note on related work tells the reader that Theorems 1.1 and 4.5 are analogous to results already obtained by Tan–Xue–Xue, and Remark 5 states plainly that Theorem 6.1 does not cover vortex patches. That is honest scholarship. It also means the novelty is narrower than the abstract might suggest.

The soft spot is real and load-bearing for Theorem 1.1. Propositions 5.4 and 5.6, which provide the logarithmic bound on ∇v and the Gronwall inequalities for the patch boundary, are stated and then delegated to [17, §8.3.3]. The only adjustment supplied is the assertion that S^(2) is L^1, contributes an additive constant to the L∞ bound of ∇v, and is absorbed into the coefficients of the Gronwall inequality. The stress-test concern lands: for the log-Gronwall mechanism to close, that additive constant has to enter without creating an extra term proportional to |∇φ|_γ in the differential inequality for |∇φ|_γ. The manuscript does not display that computation. This is a presentation gap rather than a demonstrated error — the structure of the decomposition makes the claim plausible — but for a global-in-time regularity theorem, the reader should not have to guess.

Minor: Section 3.2 global smooth existence is also a sketch. That is less concerning because the mechanism is standard and the result is not the headline.

Who this is for: researchers working on active scalar equations, vortex patches, and geophysical fluid models. The convergence theorem will be cited; the patch theorem is useful mainly as a clean statement with different methods than [22]. I would bring it to a reading group, with the explicit task of checking the Gronwall absorption. Send it to peer review, with a clear request to supply or precisely reference the missing global-regularity arguments and to adjust the introduction so the reader sees the relation to [22] upfront.

Referee Report

2 major / 3 minor

Summary. The paper studies the 2D quasi-geostrophic shallow-water (QGSW) equations, in which the velocity is obtained from the scalar vorticity via the kernel K(x) = ε/(2π) x⊥/|x| K1(ε|x|), and proves two main results. Theorem 1.1 asserts that if the initial vortex patch has boundary of class C^{1,γ}, then the unique weak solution remains a patch with C^{1,γ} boundary for all time. Theorem 6.1 asserts that, for initial data in the little Hölder space c^γ_c, the QGSW solution converges to the corresponding 2D Euler solution in C^γ norm locally in time as ε→0. The proofs use modified Bessel function estimates, Hölder-space kernel bounds, particle-trajectory methods, the contour dynamics equation, and a log-Gronwall mechanism adapted from the Euler case. The paper also proves local existence and uniqueness of regular solutions, global existence of weak solutions, and uniqueness of weak solutions in the Yudovich class.

Significance. If the results are correct, the paper fills a natural gap in the literature: previously, vortex-patch regularity for non-homogeneous kernels such as the QGSW one had not been established, and the convergence of QGSW to Euler in Hölder spaces provides a rigorous justification of the formal limit ε→0. The paper contains several concrete new estimates — for example, the ε-uniform bounds on the Bessel kernel and its derivatives in Lemma 2.2, the decomposition of ∇K into a singular part with zero spherical mean and an integrable part in Remark 1, and the commutator computation in Proposition 5.5. These are nontrivial and will be useful beyond this paper. The statements are clean, the relation to prior work is acknowledged, and the convergence proof in Section 6 is largely self-contained. The main weakness is that the global-in-time vortex-patch regularity result rests on two propositions that are stated but not proved, with only a reference to the Euler textbook argument [17, §8.3.3].

major comments (2)

[Section 5.2, Propositions 5.4 and 5.6] Theorem 1.1's global-in-time persistence relies entirely on Theorem 5.3, whose proof is not given. The text states 'The proof of Theorem 5.3 comes from combining the following results. For the details, see [17, §8.3.3]' and then proves only Proposition 5.5. Proposition 5.4 (the logarithmic bound on ||∇v||∞) and Proposition 5.6 (the Gronwall inequalities for |∇φ|_γ, ||∇φ||∞, |∇φ|_inf) are not proved. The only adjustment for the non-homogeneous QGSW kernel is the assertion after Proposition 5.6 that the L^1 part S^(2) contributes an additive constant and is absorbed into the Gronwall coefficients. This is the load-bearing point: the double-exponential control of |∇φ(t)|_γ requires that the differential inequality in Proposition 5.6 retains exactly the Euler structure after an additive constant. If S^(2) injected a term proportional to |∇φ|_γ into the inequality for |∇φ|_γ without the ||∇v|
[Section 3.2 (Global existence of smooth solutions)] Theorem 3.1 asserts global-in-time well-posedness in C^γ_c for q0∈C^γ_c, but the global half is only sketched. The paragraph after Theorem 3.6 states that the only way to leave O_M is unboundedness of ∥X∥_{1,γ}, and then asserts 'one can see that ∫ ||∇v||∞ ds is controlled by ∫ ||q||∞ ds'. This estimate alone does not control ∥X∥_{1,γ}; one also needs a uniform bound on |∇v|_γ, obtained from Lemma 2.3(25), and a Gronwall estimate using the composition bounds in Lemma 2.1(16). Without these, the global existence of the flow map is not established. This gap propagates to Theorem 4.3, where the all-time weak-solution existence is obtained by applying Theorem 3.1 to mollified initial data. The missing argument is standard, but it is load-bearing for the stated global results and should be included.

minor comments (3)

[Propositions 5.4 and 5.5] The notation v = ∇^⊥_x K0 * χ_Ω omits the ε-argument. The QGSW kernel is ∇^⊥ K0(ε|x-y|), and the statements should read accordingly to avoid confusion with the Euler kernel.
[Section 2 / Section 6] The little Hölder space c^γ_c is introduced only in Section 6, although the abstract refers to it. Consider defining it in Section 2 along with the usual Hölder spaces.
[Lemma 6.3] In the proof of Lemma 6.3, the constants in several inequalities depend on the support radius R and on the bi-Lipschitz constants of X∈U_δ. It would improve readability to state explicitly that the final estimates are uniform for X∈U_δ and q0 in a fixed bounded subset of c^γ_c.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular dependence: the central claims reduce neither to fitted inputs nor to self-citation chains; the global patch-regularity proof contains an unverified delegation gap (not a circularity).

full rationale

The derivation chain for Theorem 1.1 is not circular. Local existence and uniqueness of the CDE (Proposition 5.1) is proved in the paper from kernel expansions and a general Lipschitz-kernel lemma (Lemma 5.2). The global-in-time part is delegated to the Euler-case structure: Theorem 5.3 is assembled from Propositions 5.4–5.6, with details referred to Majda–Bertozzi [17, §8.3.3]. That is an external textbook, not a self-citation, and the QGSW-specific ingredient—the S^{(1)}+S^{(2)} decomposition of ∇K—is derived and proved in Remark 1 and Lemma 2.3. The statement that S^{(2)}, being L^1, contributes only an additive constant and is absorbed into the Gronwall coefficients is an assertion rather than a displayed computation; that is a proof-completeness risk, not a circularity, because it is not equivalent to the target conclusion and does not rename an input as a prediction. Theorem 6.1 is also not circular: the convergence proof is self-contained once well-posedness in c^γ_c is assumed; it proves uniform estimates for F^ε−F^0 (Lemma 6.3) and applies Gronwall. The self-citation [16] is used only for background well-posedness of similar active scalar equations and does not carry the convergence claim, which is compared against the known Euler equation and is independently established in the paper. No fitted parameters, no definitional dependencies, and no result that is forced by a self-referential chain were found. Score 1 reflects the minor, non-load-bearing self-citation and the otherwise external, standard references; the noted proof gap belongs to correctness risk, not circularity.

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

No free parameters. The central claims rest on standard Bessel/Calderón–Zygmund theory plus prior Euler-regime results. The main load-bearing assumption is that the non-homogeneous kernel adds only smoother terms to the Euler singularity, so the [17] global-regularity argument transfers unchanged in structure.

assumptions (5)

standard math Modified Bessel function estimates (20), asymptotics (4)–(7), and recurrences (8)–(9) from [1, Ch. 9] and [25]
Used throughout Lemmas 2.2/2.3 and Section 6 to bound the kernel and its derivatives independently of ε.
domain assumption Euler vortex patch global regularity theory from [17, §8.3.3], including the logarithmic bound and the defining-function growth estimates (Propositions 5.4, 5.6)
The QGSW global patch proof asserts these carry over once the kernel is decomposed into S^{(1)}+S^{(2)}; the proofs are not reproduced.
domain assumption Existence and continuity of the Euler solution map in little Hölder spaces c^{1,γ} (from [19] and [16])
Used in Section 6 to compare QGSW flows to the Euler flow and to obtain a uniform time T.
domain assumption Log-Lipschitz stability lemma for the QGSW kernel (Lemma 4.4, essentially [13, Lemma 2.3]) and the Lagrangian uniqueness argument of [10]
Underpins Theorem 4.5 (Yudovich uniqueness) for L∞ initial data.
standard math Cotlar's lemma (as in [23, p. 291]) and the Main Lemma of [18] for even homogeneous kernels
Used in the proof of Proposition 5.5 to bound the L∞ norm of the commutator kernel H*χ_Ω.

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Cite this review

Pith. "Pith review of The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations." pith.science (2026). https://pith.science/paper/HCVXUKHZ

@misc{pith2026260222767,
  author       = {Pith},
  title        = {Pith review of: The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCVXUKHZ}},
  note         = {Machine review of arXiv:2602.22767}
}

abstract

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $\varepsilon^{-1}$, which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW equations converge locally in time to the corresponding Euler solutions as $\varepsilon \to 0$ in little H\"older spaces.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On stationary Quasi-Geostrophic Shallow-Water flows
math.AP 2026-07 accept novelty 7.0 of 10

Non-trivial m-fold doubly-connected stationary vortex patches are proven to exist for the quasi-geostrophic shallow-water equations via bifurcation from annuli.

Reference graph

Works this paper leans on

27 extracted references · 1 linked inside Pith · cited by 1 Pith paper

[22]

A revisit of patch solutions for the 2d loglog-euler type equation.arXiv preprint arXiv:2510.18759, 2025

Changhui Tan, Liutang Xue, and Zhilong Xue. A revisit of patch solutions for the 2d loglog-euler type equation.arXiv preprint arXiv:2510.18759, 2025

arXiv 2025
[1]

Stegun, editors.Handbook of mathematical functions with formulas, graphs, and mathematical tables

Milton Abramowitz and Irene A. Stegun, editors.Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition

1992
[2]

Walter de Gruyter & Co., Berlin, 1990

Herbert Amann.Ordinary differential equations, volume 13 ofDe Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1990. An introduction to nonlinear analysis, Translated from the German by Gerhard Metzen

1990
[3]

Springer, Heidelberg, 2011

Hajer Bahouri, Jean-Yves Chemin, and Rapha¨ el Danchin.Fourier analysis and nonlinear partial differential equations, volume 343 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011

2011
[4]

Bertozzi and Peter Constantin

Andrea L. Bertozzi and Peter Constantin. Global regularity for vortex patches.Comm. Math. Phys., 152(1):19–28, 1993

1993
[5]

Bertozzi, John B

Andrea L. Bertozzi, John B. Garnett, Thomas Laurent, and Joan Verdera. The regularity of the boundary of a multidimensional aggregation patch.SIAM Journal on Mathematical Analysis, 48(6):3789–3819, 2016

2016
[6]

Bertozzi, Thomas Laurent, and Flavien L´ eger

Andrea L. Bertozzi, Thomas Laurent, and Flavien L´ eger. Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions.Math. Models Methods Appl. Sci., 22:1140005, 39, 2012

2012
[7]

Juan Carlos Cantero.C γ well-posedness of some non-linear transport equations.Indiana Univ. Math. J., 73(1):81–109, 2024

2024

Show all 27 references

[8]

The regularity of the boundary of vortex patches for some nonlinear transport equations.Analysis & PDE, 16(7):1621–1650, 2023

Juan Carlos Cantero, Joan Mateu, Joan Orobitg, and Joan Verdera. The regularity of the boundary of vortex patches for some nonlinear transport equations.Analysis & PDE, 16(7):1621–1650, 2023

2023
[9]

Persistance de structures g´ eom´ etriques dans les fluides incompressibles bidimensionnels

Jean-Yves Chemin. Persistance de structures g´ eom´ etriques dans les fluides incompressibles bidimensionnels. Ann. Sci. ´Ecole Norm. Sup. (4), 26(4):517–542, 1993

1993
[10]

An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces.Calc

Gianluca Crippa and Giorgio Stefani. An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces.Calc. Var. Partial Differential Equations, 63(7):Paper No. 168, 31, 2024

2024
[11]

DiPerna and Pierre-Louis Lions

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

1989
[12]

Dritschel, Taoufik Hmidi, and Coralie Renault

David G. Dritschel, Taoufik Hmidi, and Coralie Renault. Imperfect bifurcation for the quasi-geostrophic shallow-water equations.Arch. Ration. Mech. Anal., 231(3):1853–1915, 2019

1915
[13]

Dynamic behavior of a multilayer quasigeostrophic model: weak and time-periodic solutions.Pure Appl

Zineb Hassainia and Haroune Houamed. Dynamic behavior of a multilayer quasigeostrophic model: weak and time-periodic solutions.Pure Appl. Anal., 7(2):251–297, 2025

2025
[14]

Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations.arXiv preprint arXiv:2110.13751, 2021

Taoufik Hmidi and Emeric Roulley. Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations.arXiv preprint arXiv:2110.13751, 2021. To appear in M´ emoires de la Soci´ et´ e Math´ ematique de France, arXiv:2110.13751

2021 arXiv
[15]

Sharp strong convergence in ideal flows.Nonlinearity, 38(12):125012, dec 2025

Haroune Houamed and Marc Maga˜ na. Sharp strong convergence in ideal flows.Nonlinearity, 38(12):125012, dec 2025

2025
[16]

Continuity of the solution map to some active scalar equations in H¨ older and Zygmund spaces

Marc Maga˜ na. Continuity of the solution map to some active scalar equations in H¨ older and Zygmund spaces. Proceedings of the American Mathematical Society, 154(3):1105–1115, 2026

2026
[17]

Majda and Andrea L

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

2002
[18]

Extra cancellation of even Calder´ on-Zygmund operators and quasiconformal mappings.J

Joan Mateu, Joan Orobitg, and Joan Verdera. Extra cancellation of even Calder´ on-Zygmund operators and quasiconformal mappings.J. Math. Pures Appl. (9), 91(4):402–431, 2009

2009
[19]

Continuity of the solution map of the Euler equations in H¨ older spaces and weak norm inflation in Besov spaces.Trans

Gerard Misio lek and Tsuyoshi Yoneda. Continuity of the solution map of the Euler equations in H¨ older spaces and weak norm inflation in Besov spaces.Trans. Amer. Math. Soc., 370(7):4709–4730, 2018

2018
[20]

Dritschel

Hanna P lotka and David G. Dritschel. Quasi-geostrophic shallow-water vortex-patch equilibria and their stability.Geophys. Astrophys. Fluid Dyn., 106(6):574–595, 2012

2012
[21]

Vortex rigid motion in quasi-geostrophic shallow-water equations.Asymptot

Emeric Roulley. Vortex rigid motion in quasi-geostrophic shallow-water equations.Asymptot. Anal., 133(3):397–446, 2023

2023
[23]

Academic Press, Inc., Orlando, FL, 1986

Alberto Torchinsky.Real-variable methods in harmonic analysis, volume 123 ofPure and Applied Mathematics. Academic Press, Inc., Orlando, FL, 1986

1986
[24]

Vallis.Atmospheric and oceanic fluid dynamics: fundamentals and Large-Scale Circulation

Geoffrey K. Vallis.Atmospheric and oceanic fluid dynamics: fundamentals and Large-Scale Circulation. Cambridge University Press, 2017

2017
[25]

Watson.A Treatise on the Theory of Bessel Functions

George N. Watson.A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944

1944
[26]

Global solutions of quasi-geostrophic shallow-water fronts.J

Fangchi Yan and Qingtian Zhang. Global solutions of quasi-geostrophic shallow-water fronts.J. Differential Equations, 406:1–86, 2024

2024
[27]

Yudovich

Viktor I. Yudovich. Some bounds for solutions of elliptic equations.Translations. Series 2. American Mathematical Society., 56, 01 1963. 36 MARC MAGA ˜NA, JOAN MATEU, AND JOAN OROBITG Departament de Matem`atiques – Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelon...

1963

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [22]

A revisit of patch solutions for the 2d loglog-euler type equation.arXiv preprint arXiv:2510.18759, 2025

Changhui Tan, Liutang Xue, and Zhilong Xue. A revisit of patch solutions for the 2d loglog-euler type equation.arXiv preprint arXiv:2510.18759, 2025

arXiv 2025

[2] [1]

Stegun, editors.Handbook of mathematical functions with formulas, graphs, and mathematical tables

Milton Abramowitz and Irene A. Stegun, editors.Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition

1992

[3] [2]

Walter de Gruyter & Co., Berlin, 1990

Herbert Amann.Ordinary differential equations, volume 13 ofDe Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1990. An introduction to nonlinear analysis, Translated from the German by Gerhard Metzen

1990

[4] [3]

Springer, Heidelberg, 2011

Hajer Bahouri, Jean-Yves Chemin, and Rapha¨ el Danchin.Fourier analysis and nonlinear partial differential equations, volume 343 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011

2011

[5] [4]

Bertozzi and Peter Constantin

Andrea L. Bertozzi and Peter Constantin. Global regularity for vortex patches.Comm. Math. Phys., 152(1):19–28, 1993

1993

[6] [5]

Bertozzi, John B

Andrea L. Bertozzi, John B. Garnett, Thomas Laurent, and Joan Verdera. The regularity of the boundary of a multidimensional aggregation patch.SIAM Journal on Mathematical Analysis, 48(6):3789–3819, 2016

2016

[7] [6]

Bertozzi, Thomas Laurent, and Flavien L´ eger

Andrea L. Bertozzi, Thomas Laurent, and Flavien L´ eger. Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions.Math. Models Methods Appl. Sci., 22:1140005, 39, 2012

2012

[8] [7]

Juan Carlos Cantero.C γ well-posedness of some non-linear transport equations.Indiana Univ. Math. J., 73(1):81–109, 2024

2024