Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities

Pedro Leite; Sim\~ao Correia

REVIEW 3 major objections 4 minor 1 cited by

Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read One flexible frequency-restricted estimate at a base order and dimension implies sharp local well-posedness and nonlinear smoothing for every higher order and dimension, confirming the conjecture in [7].

desk verdict Genuinely new induction machinery with a solid gKdV application, undercut by a specific gap in the base estimates for gZK in d≥3 and NLS in d=2: σ≈±2ξ is discarded by a remark that only applies to the induction step, not the base case. read the letter →

arxiv 2412.11808 v1 pith:5IXEATFP submitted 2024-12-16 math.AP

Simão Correia , Pedro Leite This is my paper · ORCID

classification math.AP MSC 35Q5335Q5535A0135B6542B37

keywords dispersiveequationslocalwell-posednessnonlinearsmoothingfrequency-restrictedestimatesgeneralizedKdVZakharov-KuznetsovequationSchrödingermultilinear

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 induction principle for a general dispersive equation with a monomial nonlinearity of order k on R^d: if one explicit multiplier estimate, the flexible frequency-restricted estimate, holds at order k0 in dimension d0, then the equation is locally well-posed in H^s for every k>=k0 and d>=d0 whenever s is above the scaling-critical exponent, and the nonlinear part of the flow gains epsilon derivatives. The gain is exactly the quantity conjectured in [7], epsilon_c = min{(k-1)(s-s_c), ell-n-1}. The paper verifies the base estimate for the generalized KdV with k=5, for the generalized Zakharov-Kuznetsov equation with k=3 in dimensions d>=3, and for the cubic nonlinear Schrodinger equation in dimension 2, and therefore obtains sharp local well-posedness and smoothing for all higher k in those families. A reader should care because the result turns a folklore statement, namely that higher-order nonlinearities and higher dimensions improve well-posedness, into a theorem that is checked at a single base case rather than re-proved equation by equation.

What carries the argument

The load-bearing object is the flexible frequency-restricted estimate: a uniform bound with gain $M^{{1-}}$ on the measure of deformed convolution hyperplanes Gamma^sigma_xi, cut out by the resonance function Phi = L(xi) - sum L(xi_j), after frequency weights are split between two complementary sets A and A^c. The induction uses Lemma 1.9 of [7] to pass from such estimates to the multilinear $X^{{s,b}}$ estimate (1.6), together with the multiplier comparison (1.10) relating order-k multipliers to the k0 largest frequencies, and the notion of a k0-descent of the resonance function, which records which input frequencies are erased when the order is lowered. The flexibility parameter $\sigma$ is what makes the base estimate stable enough to survive the induction from k0 to larger k and from d0 to larger d.

What would settle it

Find a sign pattern in (5.1) for k=3, d=2, or a frequency region in Proposition 4.1, for which the resonance function has a degenerate critical point on Gamma^sigma_xi with $\sigma$ not excluded by Remark 2.2; then the corresponding integral (1.8) or (1.9) would fail the $M^{{1-}}$ bound, and the matching theorem would be false. Equivalently, a direct numerical check of $M^{{-1}}$ sup over $\sigma$ and $\alpha$ of the relevant integral exceeding 1 for large M would refute the paper's claim.

Watch

Extended reading notes

Core claim

The central claim is an induction principle for semilinear dispersive equations. Theorem 1.11 states: if a flexible frequency-restricted estimate (1.8)-(1.9) holds at order k0 for every k0-descent of the resonance function, then for every k>=k0, every s>s_c(k,d), and every epsilon<epsilon_c(k,d,s)=min{(k-1)(s-s_c(k,d)), ell-n-1}, the initial-value problem is locally well-posed in H^s and its flow exhibits a nonlinear smoothing effect of order epsilon. Theorem 1.12 is the analogous induction in the spatial dimension, assuming the dispersion splits as a sum of one-dimensional operators and the nonlinear multiplier is dominated by its value on the largest d0 coordinates. The paper verifies the base estimate for the quintic gKdV, for the cubic gZK in dimensions d>=3, and for the cubic NLS in dimension two, and thereby establishes sharp thresholds and explicit smoothing gains for all k>=5 gKdV, k>=5 gZK in d=2, k>=3 gZK in d>=3, and odd k>=3 NLS in d>=2. In particular, the paper confirms the conjecture made in [7] that the smoothing exponent is the minimum of the two quantities (k-1)(s-s_c) and ell-n-1.

Load-bearing premise

The whole argument rests on one unproved input: that the flexible frequency-restricted estimate holds at a single base order and dimension, for every way of lowering the resonance function. If even one of those base scenarios fails, the induction theorems have nothing to stand on.

Editorial extensions

If this is right

For the generalized KdV with k>=5, local well-posedness holds in H^s for every s>1/2-2/(k-1) and the flow gains epsilon<min{(k-1)(s-1/2+2/(k-1)),1} derivatives over the linear evolution.
For the generalized Zakharov-Kuznetsov equation, the same sharp statement holds for k>=5 in dimension 2 and for k>=3 in dimensions d>=3.
For odd-order nonlinear Schrodinger equations with k>=3 in dimension d>=2, local well-posedness holds for s>d/2-2/(k-1) and smoothing of order epsilon<min{(k-1)(s-d/2+2/(k-1)),1}; for k>=7 this improves previously known smoothing results.
Whenever the base flexible estimate holds at (k0,d0), equation (1.1) is locally well-posed above the scaling-critical regularity for all k>=k0 and d>=d0, confirming the folklore statement that higher orders and higher dimensions improve the threshold.
With the same base estimate, the nonlinearity map is bounded from X^{s,b} to X^{s+epsilon,b'} for epsilon<epsilon_c, giving a contraction argument and Lipschitz dependence of the flow on the data.

Reading between the lines

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

[Editorial extension] If a base flexible estimate were verified at k0=4 for the generalized KdV, the induction would automatically cover k>=4, the regime currently reached by separate methods; the paper stops at k0=5 because the k=4 estimate is harder.
[Editorial extension] Since the proof only needs the dispersion to split as a sum of one-dimensional operators and the multiplier to be dominated by its largest-coordinate value, the same induction should apply to other equations of this form, such as anisotropic or fractional versions, once the base estimate is checked.
[Editorial extension] The exponential-in-k growth of the bounds suggests that a single base estimate could handle infinite analytic nonlinearities for s>=d/2, turning the corresponding remark in the paper into a theorem if the coefficients decay fast enough.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 induction framework is the real news here, and it holds up on my reading: Theorem 1.11 reduces LWP and smoothing for all k≥k0 to a single flexible frequency-restricted estimate, and the weight-transfer argument in the proof is coherent. The gKdV application (Theorem 1.1) and the symmetrized d=2 gZK corollary (Theorem 1.3) look solid; Proposition 3.1 is self-contained, does not lean on the suspect remark, and gives new smoothing bounds for k≥5.

But the stress-test concern is valid, and it hits two of the three main applications. In Proposition 4.1 (gZK, d≥3, k0=3) and Proposition 5.1 (NLS, d=2, k0=3), the authors discard σ≈±2ξ by citing Remark 2.2. That remark is an induction-step device: it shows that the σ created by summing k−k0 extra frequencies avoids certain values c. In the base case k0=k there is no such sum, and (2.2) only excludes c=0. So c=±2 are not avoidable. The hyperplane condition is compatible with σ≈±2ξ when all frequencies are comparable, as long as the implicit constant in the definition of Γ^σ_ξ is larger than 2—and the induction argument itself needs such constants, since σ can be a sum of up to k−k0 frequencies. Worse, these are exactly the configurations where the Hessian degenerates: in Proposition 5.1 Case 2 the resonance is stationary along ξ with D²Φ=0, and in Proposition 4.1 the semi-nondegeneracy condition (4.3) fails. So the base estimates for gZK d≥3 and NLS d=2 are not established as written, and Theorems 1.4 and 1.5 inherit the gap.

This is a load-bearing flaw, not a cosmetic one. The fix may be near at hand—σ≈±2ξ could be treated by a direct nondegenerate change of variables in a different variable, or via the exact quadratic geometry of the NLS resonance—but it is not in the manuscript. The proofs of Propositions 4.1 and 5.1 are also sketched with references to [7] for analogous arguments; that was a minor issue before, but it compounds with the gap.

I would send this to a serious referee: the induction theorems are important enough that the community needs to see whether the base cases can be repaired. But the referee should be instructed to hold the line on Propositions 4.1 and 5.1 before acceptance. The gKdV and d=2 gZK results are publishable on their own; the authors could fix the base estimates or trim the claims to what is actually proven.

Referee Report

3 major / 4 minor

Summary. This paper develops an abstract induction framework for the local well-posedness (LWP) and nonlinear smoothing of dispersive equations of the form (1.1) with monomial nonlinearities of order k. The main results, Theorems 1.11 and 1.12, state that if a certain flexible frequency-restricted estimate holds at a base order k0 and dimension d0, then for all k >= k0 and d >= d0 one obtains LWP in H^s for s > s_c(k,d) and nonlinear smoothing of order epsilon < epsilon_c(k,d,s). The framework is applied to generalized KdV with k >= 5, generalized Zakharov-Kuznetsov with d=2, k >= 5 and with d >= 3, k >= 3, and odd NLS with d >= 2, k >= 3.

Significance. If the main theorems hold, the paper provides a unified confirmation of the monotonicity heuristic (1.2) and of the smoothing conjecture (1.3), giving sharp LWP thresholds and explicit smoothing orders in several new regimes. The proof of the abstract induction in Section 2 is coherent, and the weight bookkeeping in Theorems 1.11 and 1.12 is consistent; in particular, the induction does not presuppose the LWP statement it proves, but reduces it to a base estimate. The applications are natural and would significantly extend known results, especially nonlinear smoothing for gKdV with k >= 5 and for odd NLS with k >= 7. The paper is well organized and the flexible-estimate formalism is a valuable contribution. However, as detailed below, the verification of the base estimates in the gZK and NLS applications contains a gap that must be repaired before those applications are accepted.

major comments (3)

[Section 4, Proposition 4.1, Step 1] The exclusion of sigma approximately 2xi and sigma approximately -2xi via Remark 2.2 is not justified for the base estimate k0 = k = 3. In the notation of (2.2), for k0 = k = 3 the admissible values are c != 1 - k0/K with K = 3, so only c = 0 is excluded; c = +/-2 are in principle allowed. The proof of Remark 2.2, however, assumes that the K frequencies comparable to |xi1| are all aligned with xi, whereas the configurations listed in Step 1 involve sign cancellations, e.g. (+, -, -) and (-, -, +). Moreover, |sigma| approximately 2|xi| can be compatible with min_j |xi_j| >~ |sigma| in (1.8)-(1.9) depending on the implicit constant. As written, Proposition 4.1 does not establish the flexible estimate in the fully resonant regime, and Theorem 1.4 inherits this gap. The authors should either prove the exclusion directly for c = +/-2 or verify the estimate in these configurations.
[Section 5, Proposition 5.1, Cases 2 and 4] The same gap occurs in the NLS base estimate. Case 2 discards sigma approximately -2xi when xi1 approximately xi2 approximately -xi3, and Case 4 discards sigma approximately 2xi when xi1 approximately xi2 approximately xi3. Both exclusions are referred to Remark 2.2, which, as noted in the previous comment, does not apply at k0 = k = 3 with the required rigor. These are precisely the configurations where the direct Hessian argument degenerates (D^2_xi1 Phi = 0 in Case 2, and no nondegenerate Hessian is available in Case 4). Since (5.2)-(5.3) are the core of Proposition 5.1, Theorem 1.5 is not fully established as written.
[Section 4, Proposition 4.1, Step 2] The assertion that the semi-nondegeneracy condition rank(D^2_p1 P) >= 2 holds whenever |nabla P| << 1 and p1 approximately -p3 is stated without proof. This rank condition is load-bearing because it is what makes the Morse Splitting lemma applicable in Subcase 1.2 and yields the power M^{1-} in (4.1)-(4.2). The authors should provide a direct verification of this condition rather than leaving it as an implicit consequence of (4.3).

minor comments (4)

[Section 2, Remark 2.2] The notation sigma approximately c xi and the phrase 'avoidable values of sigma' are not defined precisely; in particular, it should be made explicit whether the comparability is in magnitude or as vectors, and what implicit constants are allowed in (1.8)-(1.9).
[Section 2, Proof of Theorem 1.12] The definition of Delta s = ((d - d0)/2)^+ is informal. Since the proof of Theorem 1.12 depends on the exact size of this shift, the authors should state a precise choice (for example, a fixed epsilon > 0 in the exponent) and verify the resulting inequalities.
[Sections 4 and 5] Several cases in Propositions 4.1 and 5.1 are dismissed as 'completely analogous' or 'similar' to earlier cases. Given that the sign combinations in the NLS resonance function (5.1) are asymmetric, the reader would benefit from a short explanation of why each indicated case reduces to the displayed prototype.
[Throughout] There are typographical and formatting issues, including 'correspondng' in Theorem 1.11 and the ambiguous use of the symbol rendered as 'fi' for both comparability and non-comparability. A consistent notation table would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the induction theorems reduce the target results to stated base multiplier estimates, which are not the same as the conclusions.

full rationale

The paper's derivation chain is a forward implication: a flexible frequency-restricted estimate (1.8)-(1.9) at (k0,d0) feeds Theorem 1.11 (induction in k) and Theorem 1.12 (induction in d), which then produce the multilinear estimate (1.6) through Lemma 1.9, yielding the stated local well-posedness and nonlinear smoothing. The base estimate is a stronger phase-multiplier condition involving deformed hyperplanes and weight splittings; it is not a restatement of the LWP/smoothing conclusion. In the proof of Theorem 1.11, the choice tilde{s} = s - Delta{s} is explicit, and the algebra showing eps < eps_c(k,d,s) follows from eps < eps_c(k0,d,tilde{s}) and the definition of sc; this is a computation, not an insertion of the conclusion. Theorem 1.12 similarly reduces the d-dimensional estimate to the d0-dimensional one by integrating out extra variables. The applications do verify base estimates directly (Proposition 3.1 for quintic gKdV, Proposition 4.1 for cubic gZK in d >= 3, Proposition 5.1 for cubic NLS in d = 2), using Morse-type phase arguments. The reliance on the authors' earlier paper [7] for Lemma 1.9 and for the sigma = 0 versions of the gZK and NLS base estimates is self-citation, but [7] contains published proofs of those ingredients and the present induction does not collapse to those results by definition. The skeptical concern about Remark 2.2 excluding sigma ≈ +/-2xi is a possible correctness gap in the proofs of Propositions 4.1 and 5.1, not a circularity: even if that gap is real, the theorem would fail for lack of proof of a hypothesis, not because the conclusion is assumed or defined into the hypotheses.

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

The central theorem is conditional on the flexible frequency-restricted estimate and on the structural assumptions (1.10)-(1.11). No numerical constants are fitted to data; the auxiliary weight-splitting exponents are proof devices, not empirical parameters. The applications supply the base estimate proofs, so the ledger entries are explicit rather than hidden.

assumptions (7)

domain assumption Bourgain space X^{s,b} and the reduction of local well-posedness and nonlinear smoothing to the multilinear estimate (1.6), via (1.4)-(1.5).
Invoked in Section 1.3 to set up the problem; standard in dispersive PDE, not proven in this paper.
standard math Lemma 1.9 from [7] (frequency-restricted estimates imply multilinear estimates).
Taken from the authors' prior SIAM paper; it is a published theorem but not reproved here, and it is the bridge used by both induction theorems.
ad hoc to paper The flexible frequency-restricted estimate (1.8)-(1.9) holds at the base order k0 and dimension d0 for all k0-descents of Phi.
This is the central hypothesis of Theorem 1.11 and must be checked application by application; Propositions 3.1, 4.1, and 5.1 provide the checks.
domain assumption Multiplier comparison (1.10): |m_k(xi_1,...,xi_k)| is bounded by |m_k0(xitilde_1,...,xitilde_k0)|, where xitilde are the k0 largest input frequencies.
Used in the proof of Theorem 1.11 to pass from the k-th nonlinearity to the k0-th; verified for the three applications, but not automatic for a general equation.
domain assumption Additive dispersion decomposition (1.11): L(xi) = sum_{j=1}^d L0(xi_j), and the multiplier depends only on the output frequency xi.
Required by the dimension-induction Theorem 1.12; satisfied by gZK after symmetrization and by NLS (with L0(xi)=xi^2), but restrictive in general.
ad hoc to paper The base critical regularity sc(k,d0) is nonnegative in the dimension induction.
Theorem 1.12 states this as an extra hypothesis; it is used to ensure stilde > 0 after subtracting dimension weights.
standard math Morse's lemma and the parametrized versions from [14] and [26] apply to the normalized resonance functions at the stationary points.
Used in Cases 3.2, 4.1, and 5.1 to replace the phase by a nondegenerate quadratic form; the paper verifies nondegeneracy in each remaining case.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities." pith.science (2026). https://pith.science/paper/5IXEATFP

@misc{pith2026241211808,
  author       = {Pith},
  title        = {Pith review of: Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5IXEATFP}},
  note         = {Machine review of arXiv:2412.11808}
}

abstract

We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local well-posedness theories. More precisely, assuming that a certain positive multiplier estimate holds at order $k_0$ and in dimension $d_0$, we prove a sharp local well-posedness result in $H^s(\mathbb{R}^d)$ for any $k\ge k_0$ and $d\ge d_0$. Moreover, we give an explicit bound on the gain of regularity observed in the difference between the linear and nonlinear solutions, confirming the conjecture made in [CorreiaOliveiraSilva24] (doi.org/10.1137/23M156923X). The result is then applied to generalized Korteweg-de Vries, Zakharov-Kuznetsov and nonlinear Schr\"odinger equations.

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

Low regularity analysis of the Zakharov--Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$
math.AP 2025-02 accept novelty 7.0 of 10

Deterministic local wellposedness on R x T at s>3/4 (or s>1/2 under a low-frequency condition), shown optimal for the bilinear/Picard method, plus probabilistic wellposedness for generic H^s data with s>-1/26.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

[7]

Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estim ates

Simão Correia, Filipe Oliveira, and Jorge Drumond Silva . Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estim ates. SIAM J. Math. Anal. , 56(4):5604–5633, 2024

work page 2024
[1]

J. L. Bona, G. Ponce, J.-C. Saut, and C. Sparber. Dispersi ve blow-up for nonlinear Schrödinger equations revisited. J. Math. Pures Appl. (9) , 102(4):782–811, 2014

work page 2014
[2]

J. L. Bona and J.-C. Saut. Dispersive blowup of solutions of generalized Korteweg-de Vries equations. J. Diﬀerential Equations, 103(1):3–57, 1993

work page 1993
[3]

Bourgain

J. Bourgain. Fourier transform restriction phenomena f or certain lattice subsets and applications to nonlinear ev olution equations. I. Schrödinger equations. Geom. Funct. Anal. , 3(2):107–156, 1993

work page 1993
[4]

Bourgain

J. Bourgain. Fourier transform restriction phenomena f or certain lattice subsets and applications to nonlinear ev olution equations. II. The KdV-equation. Geom. Funct. Anal. , 3(3):209–262, 1993

work page 1993
[5]

Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics

Thierry Cazenave. Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics . New York University, Courant Institute of Mathematical Sciences, N ew York; American Mathematical Society, Providence, RI, 2003

work page 2003
[6]

W eissler

Thierry Cazenave and Fred B. W eissler. The Cauchy proble m for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal. , 14(10):807–836, 1990

work page 1990
[8]

Burak Erdoğan, T

M. Burak Erdoğan, T. Burak Gürel, and Nikolaos Tzirakis. Smoothing for the fractional Schrödinger equation on the torus and the real line. Indiana Univ. Math. J. , 68(2):369–392, 2019

work page 2019

Show all 30 references

[9]

Burak Erdoğan and Nikolaos Tzirakis

M. Burak Erdoğan and Nikolaos Tzirakis. Global smoothin g for the periodic KdV evolution. Int. Math. Res. Not. IMRN, (20):4589–4614, 2013

2013
[10]

Burak Erdoğan and Nikolaos Tzirakis

M. Burak Erdoğan and Nikolaos Tzirakis. Dispersive partial diﬀerential equations , volume 86 of London Mathematical Society Student Texts . Cambridge University Press, Cambridge, 2016. W ellposedn ess and applications

2016
[11]

A remark on the modiﬁed Zakharov-Kuznet sov equation in three space dimensions

Axel Grünrock. A remark on the modiﬁed Zakharov-Kuznet sov equation in three space dimensions. Math. Res. Lett. , 21(1):127–131, 2014

2014
[12]

The Fourier restrict ion norm method for the Zakharov-Kuznetsov equation

Axel Grünrock and Sebastian Herr. The Fourier restrict ion norm method for the Zakharov-Kuznetsov equation. Dis- crete Contin. Dyn. Syst. , 34(5):2061–2068, 2014

2014
[13]

On the generalized zakharov-kuznetsov equation at critical regularity

Axel Grünrock. On the generalized zakharov-kuznetsov equation at critical regularity. preprint, 2015

2015
[14]

The analysis of linear partial diﬀerential operators

Lars Hörmander. The analysis of linear partial diﬀerential operators. III . Classics in Mathematics. Springer, Berlin,
[15]

Pedro Isaza, L

J. Pedro Isaza, L. Jorge Mejía, and Nikolay Tzvetkov. A s moothing eﬀect and polynomial growth of the Sobolev norms for the KP-II equation. J. Diﬀerential Equations , 220(1):1–17, 2006

2006
[16]

Polynomial bound and nonlinear smoothing for the Benjamin-Ono equation on the circle

Bradley Isom, Dionyssios Mantzavinos, Seungly Oh, and Atanas Stefanov. Polynomial bound and nonlinear smoothing for the Benjamin-Ono equation on the circle. J. Diﬀerential Equations , 297:25–46, 2021

2021
[17]

On nonlinear Schrödinger equations

Tosio Kato. On nonlinear Schrödinger equations. II. H s-solutions and unconditional well-posedness. J. Anal. Math. , 67:281–306, 1995

1995
[18]

Kenig, Gustavo Ponce, and Luis Vega

Carlos E. Kenig, Gustavo Ponce, and Luis Vega. W ell-pos edness and scattering results for the generalized Korteweg -de Vries equation via the contraction principle. Comm. Pure Appl. Math. , 46(4):527–620, 1993

1993
[19]

A smoothing property for t he L2-critical NLS equations and an application to blowup theory

Sahbi Keraani and Ana Vargas. A smoothing property for t he L2-critical NLS equations and an application to blowup theory. Ann. Inst. H. Poincaré C Anal. Non Linéaire , 26(3):745–762, 2009

2009
[20]

W ell-posedness for the Cauchy probl em of the modiﬁed Zakharov-Kuznetsov equation

Shinya Kinoshita. W ell-posedness for the Cauchy probl em of the modiﬁed Zakharov-Kuznetsov equation. Funkcial. Ekvac., 65(2):139–158, 2022

2022
[21]

Dispersive blow-up for solutions of the Zakharov-Kuznets ov equation

Felipe Linares, Ademir Pastor, and Jorge Drumond Silva . Dispersive blow-up for solutions of the Zakharov-Kuznets ov equation. Ann. Inst. H. Poincaré C Anal. Non Linéaire , 38(2):281–300, 2021

2021
[22]

Introduction to nonlinear dispersive equations

Felipe Linares and Gustavo Ponce. Introduction to nonlinear dispersive equations . Universitext. Springer, New York, second edition, 2015

2015
[23]

Felipe Linares, Gustavo Ponce, and Derek L. Smith. On th e regularity of solutions to a class of nonlinear dispersive equations. Math. Ann. , 369(1-2):797–837, 2017

2017
[24]

Felipe Linares and João P. G. Ramos. Maximal function es timates and local well-posedness for the generalized Zakharov-Kuznetsov equation. SIAM J. Math. Anal. , 53(1):914–936, 2021

2021
[25]

On the smoothing pro perties of solutions to the modiﬁed Korteweg-de Vries equation

Felipe Linares and Márcia Scialom. On the smoothing pro perties of solutions to the modiﬁed Korteweg-de Vries equation. J. Diﬀerential Equations , 106(1):141–154, 1993

1993
[26]

Critical point theory and Hamiltonian systems , volume 74 of Applied Mathematical Sciences

Jean Mawhin and Michel Willem. Critical point theory and Hamiltonian systems , volume 74 of Applied Mathematical Sciences. Springer-Verlag, New York, 1989

1989
[27]

Solutions of semilinear Schrödinger e quations in H s

Hartmut Pecher. Solutions of semilinear Schrödinger e quations in H s. Ann. Inst. H. Poincaré Phys. Théor. , 67(3):259– 296, 1997

1997
[28]

A note on the Cauchy p roblem for the 2D generalized Zakharov-Kuznetsov equations

Francis Ribaud and Stéphane Vento. A note on the Cauchy p roblem for the 2D generalized Zakharov-Kuznetsov equations. C. R. Math. Acad. Sci. Paris , 350(9-10):499–503, 2012

2012
[29]

Nonlinear dispersive equations , volume 106 of CBMS Regional Conference Series in Mathematics

Terence Tao. Nonlinear dispersive equations , volume 106 of CBMS Regional Conference Series in Mathematics . Con- ference Board of the Mathematical Sciences, W ashington, DC ; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. 15

2006
[2007]

Pseudo-diﬀerential operators, Reprint of the 1994 ed ition

1994

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [7]

Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estim ates

Simão Correia, Filipe Oliveira, and Jorge Drumond Silva . Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estim ates. SIAM J. Math. Anal. , 56(4):5604–5633, 2024

work page 2024

[2] [1]

J. L. Bona, G. Ponce, J.-C. Saut, and C. Sparber. Dispersi ve blow-up for nonlinear Schrödinger equations revisited. J. Math. Pures Appl. (9) , 102(4):782–811, 2014

work page 2014

[3] [2]

J. L. Bona and J.-C. Saut. Dispersive blowup of solutions of generalized Korteweg-de Vries equations. J. Diﬀerential Equations, 103(1):3–57, 1993

work page 1993

[4] [3]

Bourgain

J. Bourgain. Fourier transform restriction phenomena f or certain lattice subsets and applications to nonlinear ev olution equations. I. Schrödinger equations. Geom. Funct. Anal. , 3(2):107–156, 1993

work page 1993

[5] [4]

Bourgain

J. Bourgain. Fourier transform restriction phenomena f or certain lattice subsets and applications to nonlinear ev olution equations. II. The KdV-equation. Geom. Funct. Anal. , 3(3):209–262, 1993

work page 1993

[6] [5]

Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics

Thierry Cazenave. Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics . New York University, Courant Institute of Mathematical Sciences, N ew York; American Mathematical Society, Providence, RI, 2003

work page 2003

[7] [6]

W eissler

Thierry Cazenave and Fred B. W eissler. The Cauchy proble m for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal. , 14(10):807–836, 1990

work page 1990

[8] [8]

Burak Erdoğan, T

M. Burak Erdoğan, T. Burak Gürel, and Nikolaos Tzirakis. Smoothing for the fractional Schrödinger equation on the torus and the real line. Indiana Univ. Math. J. , 68(2):369–392, 2019

work page 2019