Existence of wave operators for Zakharov-Kuznetsov equation in two space dimensions

Jun-ichi Segata

REVIEW 3 major objections 3 minor 1 cited by

Existence of wave operators for Zakharov-Kuznetsov equation in two space dimensions

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that asymptotically prescribed small free solutions of the two-dimensional Zakharov–Kuznetsov equation are realized by global solutions, with the difference decaying in $H^2$ like $t^{-\alpha}$ for any $\alpha>2/3$.

desk verdict A plausible new wave-operator result for 2D quadratic ZK, with a real gap in the key bilinear estimate that needs patching before the proof is complete. read the letter →

arxiv 2507.01288 v1 pith:XTRUL3LD submitted 2025-07-02 math.AP

Jun-ichi Segata This is my paper · ORCID

classification math.AP MSC 35Q5335P25

keywords Zakharov-Kuznetsovequationwaveoperatorsscatteringspace-timeresonancemethodfinalstateproblembilineardispersiveestimatesCoifman-MeyertheoremAiryfunctiondecay

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 an existence result for wave operators for the two-dimensional Zakharov–Kuznetsov equation, a model for weakly magnetized plasma waves with a quadratic nonlinearity. In the final-state formulation, it shows that for any sufficiently small free solution $V(t)v_\infty$ prescribed at infinity, there is an actual global solution $v(t)$ that approaches it: the $H^2$-distance decays like $t^{-\alpha}$ with $\alpha>2/3$. This is nontrivial because in two dimensions the quadratic nonlinearity sits at the borderline between short-range and long-range scattering, so the natural linear decay is not integrable in time. The construction splits the solution into the free wave, a corrected bilinear profile, and a small remainder, and the main work is a bilinear dispersive estimate obtained by the space-time resonance method. The result gives the existence of wave operators for the two-dimensional Zakharov–Kuznetsov equation in the final-state framework.

What carries the argument

The key object is the bilinear correction $v_2(t)=-(\partial_{x_1}+\partial_{x_2})\int_t^\infty V(t-\tau)[(V(\tau)f)(V(\tau)g)]\,d\tau$ and the null-structure identity (3.18), which rewrites the symbol $\xi_1+\xi_2$ as $[A(\xi,\eta)\phi+B_1(\xi,\eta)\partial_{\eta_1}\phi+B_2(\xi,\eta)\partial_{\eta_2}\phi]/p(\xi,\eta)$, where $\phi$ is the phase of the oscillatory integral and $p$ is a positive quadratic denominator. This identity is what converts the derivative nonlinearity into terms that can be integrated by parts in time (the $\phi$ terms) or in frequency (the $\partial_{\eta_j}\phi$ terms). After these integrations, the proof applies the Coifman–Meyer bilinear multiplier theorem, and the linear Airy-type decay estimate for $V(t)$ supplies the $t^{-1}$ factor. The weighted norm $X$ is chosen so that the weighted derivatives produced by the integrations by parts remain bounded.

What would settle it

Choose two compactly supported Schwartz functions $f,g$ with finite $X$ norm and compute, for large $t$, the $L^2$ norm of $(\partial_{x_1}+\partial_{x_2})\int_t^\infty V(t-\tau)[(V(\tau)f)(V(\tau)g)]\,d\tau$; if it does not stay bounded by $C t^{-1}\|f\|_X\|g\|_X$, Lemma 3.1 fails. Analytically, one can check whether the multiplier $\partial_{\sigma_1}m_{sr,1,1}(\zeta+\sigma,\sigma)$ satisfies the Hörmander–Mihlin bound $|\partial^\beta m|\lesssim(|\zeta|^2+|\sigma|^2)^{-|\beta|/2}$ on the set where the denominator $p$ vanishes; a singularity there would break the integration-by-parts step.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: there exists $\varepsilon>0$ such that every $v_\infty$ with $\|v_\infty\|_X\le\varepsilon$ defines a unique global solution $v\in C(\mathbb{R};H^1(\mathbb{R}^2))$ of $\partial_t v+\partial_{x_1}^3 v+\partial_{x_2}^3 v=(\partial_{x_1}+\partial_{x_2})(v^2)$ satisfying $\|v(t)-V(t)v_\infty\|_{H^2}\lesssim \varepsilon t^{-\alpha}$ for $t>0$, with $\alpha>2/3$, and the analogous statement holds for $t<0$. Consequently $v_\infty\mapsto v(0)$ is a well-defined wave operator on the ball of radius $\varepsilon$ in the weighted space $X$. The proof writes $v=v_1+v_2+w$, where $v_1$ is the free evolution of the final state, $v_2$ is a bilinear correction that removes the quadratic interaction, and $w$ is a remainder obtained by a contraction and compactness argument; the essential difficulty is the $L^2$ estimate for $v_2$.

Load-bearing premise

The load-bearing assumption is that the bilinear dispersive estimate of Lemma 3.1 holds: the oscillatory integral defining $v_2$ is bounded in $L^2$ by $t^{-1}$ times a product of weighted norms, which depends on every multiplier produced by identity (3.18) satisfying the Hörmander–Mihlin condition and on the linear decay being exactly $t^{-1}$.

Editorial extensions

If this is right

The wave operator $v_\infty\mapsto v(0)$ is defined on the ball $\{f\in X:\|f\|_X\le\varepsilon\}$ and takes values in $H^1(\mathbb{R}^2)$.
Every sufficiently small asymptotic state in $X$ is the scattering state of a unique global solution, so the free evolution is genuinely attained at infinity with rate $t^{-\alpha}$, $\alpha>2/3$.
The same conclusion holds for $t<0$, so the scattering construction works in both time directions.
Mass and energy conservation extend the solution from a large time $T$ to all of $\mathbb{R}$, so the global solution belongs to $C(\mathbb{R};H^1)$.

Reading between the lines

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

A natural next question, not addressed in the paper, is asymptotic completeness: whether every small global solution of the two-dimensional Zakharov–Kuznetsov equation has a scattering state in $X$; the identity (3.18) may be the right tool to attempt it.
The norm $X$ is heavy, involving weights $\langle x\rangle$ and anisotropic negative derivatives up to order $-2$; one could test whether the same wave-operator statement holds with a lighter norm, which would indicate the minimal weighted regularity needed for scattering.
Because the obstruction is the borderline quadratic nonlinearity in two dimensions, the same null-structure decomposition should transfer to other two-dimensional dispersive equations with a cubic phase and a derivative quadratic nonlinearity, such as related members of the Zakharov–Kuznetsov family.

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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

The result is new and worth attention: existence of wave operators for the 2D quadratic Zakharov-Kuznetsov equation, which was open. The strategy is the standard space-time resonance method, and the algebraic identity (3.18) is the right kind of null structure. The paper also honestly notes that asymptotic completeness remains open.

What the paper does well: the reduction to the symmetric equation, the setup with v1 and v2, and the use of Coifman-Meyer multipliers are all sound in outline. The time-resonant terms are handled cleanly. Lemma 3.3 follows from Lemma 3.1 in a straightforward way.

Where the soft spots are. The proof of Lemma 3.1 has a genuine gap. In the space-resonant term Isr,1,1, after integration by parts in η1, one of the three resulting terms is bounded by ||V(τ)(x1 f)||∞ times L2 norms of V(τ)(√2 ∂_{x2}^{-1} + ∂_{x1}∂_{x2}^{-2})g. The second factor is indeed controlled by the Z norm. The first factor is the problem. Lemma 2.1 gives t^{-2/3} decay for V(t)h in L∞, and a t^{-1} decay only for |∂1|^{1/2}|∂2|^{1/2}V(t)h. Neither implies t^{-1} for V(t)(x1 f). To close that you would need a weighted estimate like || |∂1|^{-1/2}|∂2|^{-1/2} x1 f ||_{L1} ≲ ||f||_Z, which is not stated and is not obvious because of commutator singularities. The text just cites Lemma 2.1(2.3), which does not land. This is a load-bearing step: without the t^{-1} bound for v2, the bootstrap in Section 4 does not close.

There is also a smaller issue in Section 4. The squared energy inequality leads, after multiplying by t^α and taking the supremum, to a term like T^{-3α/2+1/3}||w||_{XT}^{3/2}, not T^{-α+2/3}||w||_{XT}^2. The displayed (4.3) does not follow as written. This looks repairable, but it is another place where the proof is not self-contained.

Overall: the theorem is plausible and the architecture is sensible, but the core bilinear estimate is not fully justified. The paper deserves a serious referee, and the referee should ask for a complete proof of the missing weighted decay estimate or a modification of the norm to make it true. If that gap is closed, this is a solid contribution.

Recommendation: send to peer review with a request for major revision.

Referee Report

3 major / 3 minor

Summary. The paper claims existence of wave operators for the two-dimensional Zakharov-Kuznetsov equation (1.1) in the final-state problem framework. The strategy is to represent the solution as v = v1 + v2 + w, where v1 is the free evolution of given final data and v2 is the first Picard correction. The main analytic input is a bilinear dispersive estimate (Lemma 3.1) giving an L^2 bound of size t^{-1} for the oscillatory integral defining v2, obtained by space-time resonance decomposition and the algebraic null-structure identity (3.18). From this, Lemma 3.3 derives decay of v2 in H^3, and a bootstrap in Section 4 closes at rate t^{-α} with α > 2/3, yielding Theorem 1.1.

Significance. If the proof were complete, the result would be a significant advance: it would establish scattering for the borderline quadratic Zakharov-Kuznetsov equation in two dimensions, where naive L^∞ decay t^{-2/3} is not integrable. The paper is self-contained relative to classical tools (Coifman-Meyer, Keel-Tao Strichartz, Airy decay), and the algebraic identity (3.18) is explicit and checkable. The main concern is that a load-bearing estimate in Lemma 3.1 is not justified; if that gap is repaired, the overall strategy is plausible and the paper would merit publication.

major comments (3)

[Section 3, proof of Lemma 3.1, space-resonant term Isr,1,1] After integration by parts in η1, the second displayed summand is bounded using ||V(τ)(x1 f)||_{L∞} ≤ C τ^{-1} ||f||_Z. This bound is not a consequence of Lemma 2.1 as stated. Lemma 2.1(2.3) gives τ^{-1} decay only for |∂1|^{1/2}|∂2|^{1/2} V(τ)h with an L1 norm of h; applying it with h = |∂1|^{-1/2}|∂2|^{-1/2} x1 f would require the estimate || |∂1|^{-1/2}|∂2|^{-1/2} x1 f ||_{L1} ≲ ||f||_Z. The paper neither states nor proves such a weighted commutator estimate, and it is not automatic because x1 does not commute with the half-wave Fourier multipliers and the commutator is more singular, not less. Since (3.1) is the key input to Lemma 3.3(3.23) and to the bootstrap estimates (4.6)-(4.8), this gap is load-bearing for Theorem 1.1.
[Section 3, Lemma 3.1, remaining space-resonant terms] The proof says that all other space-resonant terms "can be treated in a similar way." Several of those terms will produce the same problematic factor V(τ)(x1 f) or analogous weighted commutator expressions after integration by parts. The author should verify explicitly that every term either avoids this factor or is controlled by the stated Z norm, or provide a general lemma covering all such cases. As written, the t^{-1} estimate for Isr is not established.
[Section 4, bootstrap estimates (4.6)-(4.8)] The closing of the bootstrap for w at rate α > 2/3 depends critically on the t^{-1} decay of v2 in H^3, which in turn rests on Lemma 3.1. If Lemma 3.1 only yielded t^{-2/3} decay for v2 (the generic linear decay rate available from Lemma 2.1), then the term in (4.6) involving ||v2||_{H^2} ||w||_{H^2} would produce t^{-α+1/3} after integration, which is not bounded by t^{-α}. Thus the missing estimate in Lemma 3.1 is not a technicality but a necessary ingredient for the stated rate.

minor comments (3)

[Section 3, proof of Lemma 3.1] In the display after "Hence Lemma 3.2 yields," the notation [V(t)f] and [V(t)(...)] appears inside integrals over τ; these should be V(τ), not V(t). The same typo occurs in the subsequent estimates for Isr,1,1.
[Section 1, references] The citation to "Kinoshita-Correia" with footnote "work in preparation" is not a complete reference. If the paper is available, it should be cited properly; if it is not used as a load-bearing input, it could be removed.
[Section 3, equation (3.16)] In the definition of B1,1, the factorization (2η2^2 − η1^2)/η2^2 = (√2η2 − η1)(√2η2 + η1)/η2^2 is used implicitly; writing the intermediate step would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is proved from independent linear dispersive estimates, an algebraic null-structure identity, and external Coifman–Meyer and Strichartz theorems; no fitted parameter, definitional identity, or load-bearing self-citation is used.

full rationale

The paper's derivation chain is self-contained and does not reduce to its own inputs. Lemma 2.1 is proved from the explicit Airy kernel decay and interpolation, with Strichartz estimates quoted from Keel–Tao. Lemma 3.1 is the key bilinear estimate; its proof uses the algebraic identity (3.18) to decompose the oscillatory integral into time-resonant and space-resonant pieces, then applies the external Coifman–Meyer multiplier theorem and the decay estimates of Lemma 2.1. No norm is fitted to the conclusion: the X and Z norms are explicitly defined weighted and negative-derivative spaces, and Theorem 1.1 asserts a genuine existence and decay statement for data in that space. The wave operator is defined directly from the constructed scattering solutions, so its existence is not assumed or renamed. The only citation to work in preparation (Kinoshita–Correia) appears in the literature review and is not used in any estimate. The one point that may be under-justified is the bound for V(tau)(x1 f) in the space-resonant term Isr,1,1: Lemma 2.1 as stated gives t^{-2/3} decay for V(t)h in L^infty and t^{-1} for |D1|^{1/2}|D2|^{1/2}V(t)h, and the paper does not explicitly show how this yields the needed t^{-1} bound for V(tau)(x1 f). That is a possible correctness gap in a lemma, but it is not circularity: the missing estimate is not obtained by defining the norm to equal the bound, nor is it imported from the author's own prior work. The central claim therefore has independent mathematical content and the paper should receive a circularity score of 0.

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

The proof rests on standard mathematical tools (Airy estimates, Coifman-Meyer, Keel-Tao) and one asserted regularity statement for a regularized equation. No parameters are fitted and no new physical or mathematical entities are postulated.

assumptions (4)

standard math Airy function decay bounds: |A(t,y)| less than or similar to t^{-1/3} and ||partial_y|^{1/2} A(t,y)| less than or similar to t^{-1/2}.
Used in Lemma 2.1 to prove the L^p decay estimates for the linear group V(t). Cites Linares-Ponce [17, Section 1.5].
standard math Coifman-Meyer bilinear multiplier theorem (Lemma 3.2).
Used throughout the proof of Lemma 3.1 to bound the decomposed bilinear oscillatory integrals. Cites Coifman-Meyer [3] and Grafakos [8, Theorem 7.5.3].
standard math Keel-Tao endpoint Strichartz estimates applied to the Airy-type group.
Used in Lemma 2.1(ii) to obtain the Strichartz inequality (2.4) for the linear flow. Cites Keel-Tao [13, Theorem 1.2].
domain assumption Global existence for the regularized equation (4.2) without smallness of v_infty.
Asserted in Section 4 as 'easily seen' using the regularizing factor rho_nu and the time-decaying factor (1 + lambda t)^{-5}; no proof is given. This is needed to set up the a priori estimates and compactness argument.

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Pith. "Pith review of Existence of wave operators for Zakharov-Kuznetsov equation in two space dimensions." pith.science (2026). https://pith.science/paper/XTRUL3LD

@misc{pith2026250701288,
  author       = {Pith},
  title        = {Pith review of: Existence of wave operators for Zakharov-Kuznetsov equation in two space dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XTRUL3LD}},
  note         = {Machine review of arXiv:2507.01288}
}

read the original abstract

In this paper, we study long time behavior of solution to the two dimensional Zakharov-Kuznetsov equation in the framework of the final state problem. We construct a small global solution to the Zakharov-Kuznetsov equation which scatters to a given free solution. From this result, we have the existence of wave operators for the Zakharov-Kuznetsov equation. The proof is based on the space-time resonance method developed by Gustafson-Nakanishi-Tsai and Germain-Masmoudi-Shatah etc.

Discussion (0). Continue with ORCID to comment.

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

Global well-posedness and scattering for the 2D modified Zakharov-Kuznetsov equation
math.AP 2025-07 accept novelty 7.0 of 10

Global well-posedness and scattering for the 2D modified Zakharov-Kuznetsov equation at a new critical regularity, achieved with an anisotropic two-parameter Sobolev space.

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Works this paper leans on

22 extracted references · 21 canonical work pages · cited by 1 Pith paper

[1]

available at arXiv:2506.17179

Anjolras P., Scattering of the 2D modified Zakharov-Kuznetsov equation , preprint. available at arXiv:2506.17179

arXiv
[2]

and Saut J.-C., Dispersion estimates for third order equations in two dimensions

Ben-Artzi M., Koch H. and Saut J.-C., Dispersion estimates for third order equations in two dimensions . Comm. Partial Differential Equations 28 (2003), 1943–1974

work page 2003
[3]

Coifman R. R. and Meyer Y., Commutateurs d’integrales singuli` eres et op´ erateurs multilin´ eaires. Ann. Inst. Fourier (Grenoble) 28 (1978), 177–202

work page 1978
[4]

V., The Cauchy problem for the Zakharov-Kuznetsov equation

Faminskii A. V., The Cauchy problem for the Zakharov-Kuznetsov equation . Differ. Equations, 31 (1995), 1002–1012

work page 1995
[5]

and Pastor A., A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results

Farah L.G., Linares F. and Pastor A., A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results . J. Differential Equations 253 (2012), 2558–2571

work page 2012
[6]

and Shatah J., Global solutions for 3D quadratic Schr¨ odinger equations

Germain P, Masmoudi N. and Shatah J., Global solutions for 3D quadratic Schr¨ odinger equations. Int. Math. Res. Not. 2009 (2009), 414–432

work page 2009
[7]

and Shatah J., Global solutions for 2D quadratic Schr¨ odinger equations

Germain P, Masmoudi N. and Shatah J., Global solutions for 2D quadratic Schr¨ odinger equations. J. Math. Pures Appl. (9) 97 (2012), 505–543

work page 2012
[8]

Grafakos L., Modern Fourier analysis. Grad. Texts in Math., 250 Springer, New York (2009)

work page 2009

Show all 22 references

[9]

and Herr S.,The Fourier restriction norm method for the Zakharov-Kuznetsov equation

Gr¨ unrock, A. and Herr S.,The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete Contin. Dyn. Syst. 34 (2014), 2061–2068

2014
[10]

Gustafson S., Nakanishi K., and Tsai T.-P., Scattering theory for the Gross-Pitaevskii equation in three dimensions. Commun. Contemp. Math. 11 (2009), 657–707

2009
[11]

and Kinoshita S., The Zakharov-Kuznetsov equation in high dimensions: small initial data of critical regularity

Herr S. and Kinoshita S., The Zakharov-Kuznetsov equation in high dimensions: small initial data of critical regularity. J. Evol. Equ. 21 (2021), 2105–2121

2021
[12]

and Kinoshita S., Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher

Herr S. and Kinoshita S., Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher . Ann. Inst. Fourier (Grenoble) 73 (2023), 1203–1267. ZAKHAROV-KUZNETSOV EQUATION IN 2D 15

2023
[13]

and Tao T., Endpoint Strichartz estimates

Keel M. and Tao T., Endpoint Strichartz estimates . Amer. J. Math. 120 (1998), 955–980

1998
[14]

Kinoshita S., Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D . Ann. Inst. H. Poincar´ e C Anal. Non Lin´ e aire38 (2021), 451–505

2021
[15]

Laedke E. W. and Spatschek K.-H. , Nonlinear ion-acoustic waves in weak magnetic fields . Phys. Fluids, 25 (1982), 985–989

1982
[16]

and Saut J.-C., The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation

Lannes D., Linares F. and Saut J.-C., The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation. Studies in phase space analysis with applications to PDEs, Progr. Nonlinear Differential Equations Appl., 84 (2013), 181–213

2013
[17]

and Ponce G., Introduction to nonlinear dispersive equations

Linares F. and Ponce G., Introduction to nonlinear dispersive equations . Universitext, Springer, New York, (2009)

2009
[18]

and Saut J.-C., The Cauchy problem for the 3D Zakharov-Kuznetsov equation

Linares F. and Saut J.-C., The Cauchy problem for the 3D Zakharov-Kuznetsov equation . Discrete Contin. Dyn. Syst., 24 (2009), 547–565

2009
[19]

and Pilod D., Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications

Molinet L. and Pilod D., Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications . Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire32 (2015), 347–371

2015
[20]

and Tsutsumi Y., Global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions , Adv

Ozawa T. and Tsutsumi Y., Global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions , Adv. Math. Sci. Appl. 3 (1993/94), Special Issue, 301–334

1993
[21]

and Vento S., Well-posedness results for the three-dimensional Zakharov-Kuznetsov equation

Ribaud F. and Vento S., Well-posedness results for the three-dimensional Zakharov-Kuznetsov equation. SIAM J. Math. Anal., 44 (2012), 2289–2304

2012
[22]

Zakharov V. E. and Kuznetsov E. A., Three-dimensional solitons. Sov. Phys. JETP, 39 (1974), 285–286. F aculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan Email address : segata@math.kyushu-u.ac.jp

1974

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

available at arXiv:2506.17179

Anjolras P., Scattering of the 2D modified Zakharov-Kuznetsov equation , preprint. available at arXiv:2506.17179

arXiv

[2] [2]

and Saut J.-C., Dispersion estimates for third order equations in two dimensions

Ben-Artzi M., Koch H. and Saut J.-C., Dispersion estimates for third order equations in two dimensions . Comm. Partial Differential Equations 28 (2003), 1943–1974

work page 2003

[3] [3]

Coifman R. R. and Meyer Y., Commutateurs d’integrales singuli` eres et op´ erateurs multilin´ eaires. Ann. Inst. Fourier (Grenoble) 28 (1978), 177–202

work page 1978

[4] [4]

V., The Cauchy problem for the Zakharov-Kuznetsov equation

Faminskii A. V., The Cauchy problem for the Zakharov-Kuznetsov equation . Differ. Equations, 31 (1995), 1002–1012

work page 1995

[5] [5]

and Pastor A., A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results

Farah L.G., Linares F. and Pastor A., A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results . J. Differential Equations 253 (2012), 2558–2571

work page 2012

[6] [6]

and Shatah J., Global solutions for 3D quadratic Schr¨ odinger equations

Germain P, Masmoudi N. and Shatah J., Global solutions for 3D quadratic Schr¨ odinger equations. Int. Math. Res. Not. 2009 (2009), 414–432

work page 2009

[7] [7]

and Shatah J., Global solutions for 2D quadratic Schr¨ odinger equations

Germain P, Masmoudi N. and Shatah J., Global solutions for 2D quadratic Schr¨ odinger equations. J. Math. Pures Appl. (9) 97 (2012), 505–543

work page 2012

[8] [8]

Grafakos L., Modern Fourier analysis. Grad. Texts in Math., 250 Springer, New York (2009)

work page 2009