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On restricted-type Strichartz estimates and the applications

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Pith's one-line read The paper establishes global sharp shell-type Strichartz estimates on R²×T with no derivative loss, constructs a counterexample forcing derivative loss on R×T², and uses these to prove local well-posedness of the Zakharov system on waveguid

desk verdict The no-loss R2×T shell estimate is a plausible and interesting dichotomy, but the paper has a gap in Theorem 1.3 and an unsupported inhomogeneous Strichartz estimate in Section 7. read the letter →

arxiv 2508.18827 v1 pith:V3QX7TR5 submitted 2025-08-26 math.AP

Yangkendi Deng , Han Wang , Yuzhao Wang , Zehua Zhao This is my paper
classification math.AP MSC 35Q5535R0137K0637L50
keywords Zakharovsystemshell-typeStrichartzestimatewaveguidemanifoldderivativelosssemi-algebraicsetmeasurelocalwell-posednesssupercriticalNLS
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 asks when a Schrödinger evolution with initial data supported on a thin spherical shell satisfies the sharp Strichartz bound without losing derivatives, and answers it for hybrid spaces R^m×T^n. The main analytic claim is that on R²×T the L⁴ spacetime estimate is global and derivative-free, proved by controlling the measure of semi-algebraic sets sliced by the integer lattice; on R×T² the analogous sharp estimate is false, with at least logarithmic loss. These shell-type estimates are the analytic engine for the paper's PDE application: a first local well-posedness theorem for the Zakharov system on waveguides. A parallel strip-type estimate yields local well-posedness for supercritical NLS with frequency strips, a deterministic analogue of random-data well-posedness.

What carries the argument

The carrying mechanism is the shell-type Strichartz estimate for the linear Schrödinger flow, i.e. an L²→L^p estimate for data whose Fourier support lies in a unit-width spherical shell of large radius. In the R²×T case the proof's engine is Lemma 4.2, a measure estimate for bounded semi-algebraic sets—sets defined by finitely many polynomial equalities and inequalities—in R²×Z: the sum over lattice slices n∈Z of the area of U∩(R²×{n}) is controlled by the Lebesgue measure |U| plus the largest slice. This turns the quadratic-resonance geometry |(ξ−a)·(ξ−b)|≤1/T into the required 1/T bound. The counterexample on R×T² uses a product of a one-dimensional Euclidean wave packet with a high-freque

What would settle it

Run the linear Schrödinger evolution on R²×T for a sequence of initial data whose Fourier transform is 1 on the full thin shell {c*−1≤|(ξ,n)|≤c*+1} (no additional ball cut-off), with c*→∞, normalize in L², and measure the L⁴_{t,x}([0,T]×R²×T) norm for a fixed finite T and also as T→∞. If the ratio grows with c* (or with T), Theorem 1.3 as stated is false or requires the missing localization step; if it stays bounded, the full-shell no-loss claim is confirmed. The same test on R×T² should show unbounded growth, matching the paper's counterexample.

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Extended reading notes

Core claim

The central discovery is a dichotomy in three-dimensional product geometries: the shell-type estimate ‖e^{itΔ}φ‖_{L⁴_{t,x}([0,∞)×R²×T)} ≲ ‖φ‖_{L²} holds for Fourier-supported data on a unit-thickness shell of radius c*, with no derivative loss and globally in time; the same family of estimates on R×T² fails at the sharp exponent, and the counterexample forces a logarithmic derivative loss. The proof of the positive result works by reducing the L²→L⁴ estimate to a measure estimate and importing a lemma on semi-algebraic sets in R²×Z: the sum of lattice-slice measures is bounded by the total measure plus the largest slice, because the slice function changes monotonicity only O(1) times. From t

Load-bearing premise

The proof of the no-loss estimate on R²×T rests on a semi-algebraic measure lemma from [4] saying that lattice-slice sums of a bounded-complexity set are controlled by its volume plus its largest slice; if that monotonicity lemma fails for the quadratic shell sets, or if the reduction from the actually-proved small-ball support to the stated full-shell support cannot be supplied, the dichotomy collapses.

Editorial extensions

If this is right

  • On R²×T, shell-supported initial data obey a global L⁴ spacetime bound with no derivative loss, so shell frequencies do not degrade the evolution's integrability over infinite time.
  • On R×T^{d−1} (including R×T²), the sharp shell estimate fails at the endpoint p = 2(d+1)/(d−1), and the constructed data show at least a logarithmic loss.
  • For general R^m×T^n, shell-type bounds hold locally in time with arbitrary ε-derivative loss via decoupling, so the no-loss phenomenon is, as far as these results go, special to R²×T.
  • The Zakharov system on R^m×T^n is locally well-posed for s > s0 (1/2 in d=3, 3/4 in d=4, (d−3)/2 for d≥5), so the shell estimates unlock a PDE application.
  • Strip-restricted initial data satisfy the full Strichartz range, giving local well-posedness for 2D quintic NLS in H^{1/8}, below the usual H^{1/2} critical regularity.

Reading between the lines

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

  • The dichotomy suggests a general 'dimension deficit' principle: when the Euclidean part has codimension at least 2, dispersion suppresses derivative loss; when it has codimension 1, loss is inevitable. This could be tested on R^m×T^n with m≥2, n≥1 beyond the known cases.
  • If shell-type estimates could be proved in the full Strichartz range (as strip-type are here), the supercritical NLS well-posedness argument would transfer verbatim to shell-restricted data, giving another deterministic analogue of random-data theory.
  • The numerical scaling hierarchy O(1), N^{0.18}, N^{0.30} hints that the optimal loss on R×T² might be logarithmic with a small power-law artifact; a sharp counterexample refined to log-loss would settle the exact constant.
  • On the PDE side, the same shell estimates should support scattering or long-time results for the Zakharov system on waveguides once global-in-time wave estimates in the waveguide setting are established.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies L^2-to-L^p Strichartz estimates for the Schrödinger evolution on frequency shells in Euclidean space and on waveguide manifolds R^m×T^n, and applies them to the Zakharov system and to supercritical NLS. Theorem 1.1 gives a global shell-type estimate on R^d with no derivative loss via Stein–Tomas. Theorem 1.3 claims a global, derivative-free L^4 shell estimate on R^2×T, proved by reducing to a measure estimate for semi-algebraic sets; Theorem 1.4 gives a counterexample on R×T^{d-1} showing logarithmic loss. Theorem 1.5 proves local shell estimates with ε-loss on general waveguides via decoupling. Theorem 1.7 claims local well-posedness of the partially periodic Zakharov system. Section 7 claims well-posedness for frequency-restricted supercritical NLS. Numerical experiments are reported in Section 9.

Significance. If the proofs were complete, the paper would make a useful contribution: the Euclidean shell estimate is a clean Stein–Tomas argument, the R^2×T versus R×T^2 dichotomy is an interesting and testable phenomenon, the counterexample in Theorem 1.4 is explicit, and a well-posedness theory for Zakharov on waveguides would be new. The use of semi-algebraic measure estimates from [4] is a promising technique. However, the manuscript as written has serious gaps in load-bearing arguments: the cap-localized proof of Theorem 1.3 is not reduced to the stated shell statement, the contraction proof in Theorem 7.5 relies on a false inhomogeneous Strichartz estimate, and the key bilinear Proposition 6.1 is asserted without proof. These issues must be resolved before the central claims can be accepted.

major comments (3)
  1. [§4.1, proof of Theorem 1.3] The proof begins by saying it suffices to prove the estimate for ϕ with supp φ̂ contained in the shell intersected with B_{c*/100}, but Theorem 1.3 contains no such ball. The reduction is attributed to a 'standard argument' from [13] and is not shown. A shell of radius c* and width 2 cannot be covered by O(1) balls of radius c*/100; a naive partition would introduce a c*-dependent factor. Without a detailed proof that the cap-localized L^4 bound implies the global shell bound, Theorem 1.3 is not established.
  2. [§7.1, Lemma 7.4 and Theorem 7.5] Lemma 7.4 is false as stated. For d=2, (q,r)=(8,8) and (q̃,r̃)=(8/3,8/3) satisfy the displayed scaling condition, but (8,8) is not Schrödinger-admissible in dimension 2. The free evolution obeys ∥e^{iτΔ}h∥_{L^8_x} ≲ τ^{-3/4}∥h∥_{L^{8/7}_x}; convolving with L^{8/5}_t gives only L^{8/3}_t by Hardy–Littlewood–Sobolev, not L^8_t. Since the contraction proof of Theorem 7.5 applies this lemma to F=|u|^4u, the estimate of the Duhamel term is unsupported. Moreover, F is not strip-frequency supported, so Theorem 7.2 cannot be used on the nonlinear term. This section needs either a correct inhomogeneous estimate or removal.
  3. [§6, Proposition 6.1] The proof of Proposition 6.1, which is the key bilinear estimate behind Theorem 1.7, is reduced to a case-by-case analysis and then 'we omit the proof'. The waveguide adaptation from the periodic setting in [32] is not a purely cosmetic change: it must use the shell-type estimates of Theorem 1.5 and the waveguide wave-equation Strichartz estimates. Without a written proof, or at least a precise statement of the modified cases and how the new estimates enter, Theorem 1.7 rests on an unproved assertion.
minor comments (4)
  1. [§4.2, proof of Theorem 1.4] The constructed data has n_j ranging up to ~N2^{1/2}. For the support to lie in the shell |(ξ,n)|∈[N1−1,N1+1], one needs N2≪N1. The statement should specify a relation such as N1=N2^2; otherwise the displayed support inclusion is not guaranteed.
  2. [§9, numerical experiments] The scaling exponents are extracted from a log-log regression over only N∈[4,32]. This is a very narrow range; please report the fitted uncertainties and, ideally, provide the codes or data for reproducibility.
  3. [§8, Tables 4 and 5] The summary tables are useful but should distinguish clearly between 'no derivative loss for R^2×T' and the ε-loss results for general waveguides. Table 4 currently says 'no loss for R^2×T' but Theorem 1.5 is local and has ε-loss; the text explains this, but the table alone is ambiguous.
  4. [Throughout] There are several typographical errors, e.g. 'of of' in Remark 1.2 and 'F unction' in the Notation section; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: main theorems derive from independent external input (Stein-Tomas, decoupling, semi-algebraic measure estimates), and no fitted value is renamed as a prediction.

full rationale

The derivation chain is self-contained relative to external results. Theorem 1.1 is proved from Stein-Tomas restriction and spherical coordinates. Theorem 1.3 reduces to the measure estimate for semi-algebraic sets (Lemma 4.2), whose proof rests on Lemma 2.9 of Basu-Guo-Zhang-Zorin-Kranich [4] via Lemmas 4.3-4.4; this is an independent external input, not an author-imported ansatz. Theorem 1.5 follows from the decoupling inequality of Kinoshita-Nakamura-Sanwal [32]. The Zakharov well-posedness (Theorem 1.7) is a standard contraction argument following [32], with Proposition 6.1's proof omitted but claimed to adapt [32]; omission is a rigor gap, not a circular reduction. The strip-type results in Section 7 use Keel-Tao and Foschi's inhomogeneous Strichartz estimates; the scaling condition is checked, and the supercriticality inequality s < s_c is verified directly from the formula. Numerical Section 9 is post-hoc: the observed scaling exponents are empirical fits and are not inputs to any theorem. Self-citations [15,16,17,20] appear only in background/future work and do not carry the proofs. There is no fitted parameter renamed as a prediction and no uniqueness theorem imported from the authors. Two non-circular correctness concerns should be flagged: Theorem 1.3's proof establishes only the shell ∩ B_{c*/100} localization and does not show how to pass to the stated shell-only support; and the application of Lemma 7.4 in Theorem 7.5 with (q,r)=(8,8), (q̃,r̃)=(8/3,8/3) is outside the standard admissible range of Foschi's inhomogeneous estimates, so the contraction proof has an unsupported step. These are mathematical gaps, not instances of circular reasoning.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central estimates rest on independent external machinery (Stein-Tomas, decoupling, semi-algebraic measure estimates) and on two unstated reductions (ball localization, standard argument from [13]). The numerical scaling exponents are fitted to simulations and play no role in the proofs.

free parameters (3)
  • Empirical scaling exponent α for R2×T = 0
    Observed O(1) scaling in Section 9, Table 7; fitted via log-log regression, not derived from theory.
  • Empirical scaling exponent α for R×T2 = 0.18
    Fitted by linear regression on log-log plot for N∈[4,32] in Section 9; Theorem 1.4 only proves failure, not this exponent.
  • Empirical scaling exponent α for T3 = 0.30
    Fitted value reported in Section 9, Table 7; used as 'quantitative confirmation' but no error bars or code are provided.
assumptions (7)
  • standard math Stein-Tomas restriction theorem on S^{d-1}
    Used in Eq (3.4) of Section 3 to prove Theorem 1.1.
  • standard math Lemma 4.4 from [4]: semi-algebraic slice functions change monotonicity O(1) times
    Imported to prove Lemma 4.2, which powers Theorem 1.3; not proved in this paper.
  • standard math Decoupling inequality (Corollary 3.3 in [32])
    Used in Section 5 to derive the ε-loss shell estimates for R^m×T^n; quoted without proof.
  • domain assumption Wave Strichartz estimates on periodic/waveguide spaces follow from Euclidean by finite speed
    Remark 2.5; used in the Zakharov proof.
  • domain assumption Shell/strip frequency support restriction on initial data
    All main theorems are conditional on this restriction; the paper does not claim unrestricted results.
  • ad hoc to paper Localization to ball B_{c*/100} in Theorem 1.3 proof
    The proof treats support in shell ∩ B_{c*/100}; the theorem as stated has no such ball.
  • ad hoc to paper The 'standard argument' from Proposition 3.7 of [13] converting the L^4 bound to a slicing measure estimate
    Invoked in the proof of Theorem 1.3 without reproduction; load-bearing.

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Pith. "Pith review of On restricted-type Strichartz estimates and the applications." pith.science (2026). https://pith.science/paper/V3QX7TR5

@misc{pith2026250818827,
  author       = {Pith},
  title        = {Pith review of: On restricted-type Strichartz estimates and the applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3QX7TR5}},
  note         = {Machine review of arXiv:2508.18827}
}
abstract

We establish a rigorous framework for the Zakharov system on waveguide manifolds $\mathbb{R}^m \times \mathbb{T}^n$ ($m,n\geq 1$), which models the nonlinear coupling between optical and acoustic modes in confined geometries such as optical fibers. Our analysis reveals that the sharp \textit{shell-type Strichartz estimate} for $\mathbb{R}^2 \times \mathbb{T}$ is globally valid in time and exhibits no derivative loss via the measure estimate of semi-algebraic sets, unlike the periodic case studied in \cite{MR4665720}. In addition, we demonstrate that such an estimate fails on the product space $\mathbb{R} \times \mathbb{T}^2$ by constructing a counter-example. Moreover, we derive analogues of these shell-type estimates in other dimensions, both in the waveguide and Euclidean settings. As a direct application, we establish, for the first time, a local well-posedness theory for the partially periodic Zakharov system. To summarize, we compare shell-type Strichartz estimates in different settings (the Euclidean, the periodic, and the waveguide). Numerical verification on $\mathbb{R}^2\times\mathbb{T}$ reveals a uniform $L^4$-spacetime bound, while $\mathbb{R}\times\mathbb{T}^2$ exhibits sublinear growth, quantitatively confirming the theoretical dichotomy between geometries with different dimensional confinement. These findings advance the understanding of dispersive effects in hybrid geometries and provide mathematical foundations for efficient waveguide design and signal transmission. Finally, for the Euclidean case, we establish well-posedness theory for supercritical nonlinear Schr\"odinger equation (NLS) with \textit{strip-type} frequency-restricted initial data, revealing a trade-off between dispersion and confinement, which is of independent mathematical interest. This provides a deterministic analogue to random data theory of NLS.

Figures

Figures reproduced from arXiv: 2508.18827 by the authors.

Figure 1
Figure 1. Log-log plot of ∥e it∆u0∥L4 /∥u0∥L2 vs. N for N ∈ [4, 32]. These observations suggest a general principle: the derivative loss in Strichartz estimates is controlled by the dimension deficit between the Euclidean and periodic components. This provides guidance for designing waveguide structures in photonic applications where dispersion properties are crucial. Open questions include what the optimal derivative losses … view at source ↗

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Works this paper leans on

45 extracted references · 43 canonical work pages · cited by 1 Pith paper

  1. [13]

    De Silva, N

    D. De Silva, N. Pavlovi´ c, G. Staffilani, and N. Tzirakis. Global well-posedness for a periodic nonlinear Schr¨ odinger equation in 1D and 2D.Discrete Contin. Dyn. Syst. , 19(1):37–65, 2007

    work page 2007

  2. [32]

    Kinoshita, S

    S. Kinoshita, S. Nakamura, and A. Sanwal. Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system. Comm. Math. Phys. , 404(2):883–926, 2023

    work page 2023

  3. [4]

    S. Basu, S. Guo, R. Zhang, and P. Zorin-Kranich. A stationary set method for estimating oscillatory integrals. arXiv preprint arXiv:2103.08844 , 2021

    work page Pith review arXiv 2021

  4. [1]

    Antonelli, R

    P. Antonelli, R. Carles, and J. D. Silva. Scattering for nonlinear Schr¨ odinger equation under partial harmonic confinement. Comm. Math. Phys. , 334(1):367–396, 2015

    work page 2015

  5. [2]

    A. Barron. On global-in-time Strichartz estimates for the semiperiodic Schr¨ odinger equation. Anal. PDE, 14(4):1125–1152, 2021

    work page 2021

  6. [3]

    Barron, M

    A. Barron, M. Christ, and B. Pausader. Global endpoint Strichartz estimates for Schr¨ odinger equations on the cylinder R × T. Nonlinear Anal., 206:Paper No. 112172, 7, 2021

    work page 2021

  7. [5]

    Bejenaru, S

    I. Bejenaru, S. Herr, J. Holmer, and D. Tataru. On the 2D Zakharov system with L2 Schr¨ odinger data.Nonlinearity, 22(5):1063–1089, 2009

    work page 2009

  8. [6]

    Bourgain

    J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applica- tions to nonlinear evolution equations. I. Schr¨ odinger equations.Geom. Funct. Anal, 3(2):107– 156, 1993

    work page 1993

Show all 45 references
  1. [7]

    Bourgain

    J. Bourgain. Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Israel J. Math. , 193(1):441–458, 2013

    2013

  2. [8]

    Bourgain and J

    J. Bourgain and J. Colliander. On wellposedness of the Zakharov system. Internat. Math. Res. Notices, 1996(11):515–546, 1996

    1996

  3. [9]

    Bourgain and C

    J. Bourgain and C. Demeter. The proof of the l2 decoupling conjecture. Ann. of Math. (2) , 182(1):351–389, 2015

    2015

  4. [10]

    Bourgain and L

    J. Bourgain and L. Guth. Bounds on oscillatory integral operators based on multilinear esti- mates. Geom. Funct. Anal., 21(6):1239–1295, 2011

    2011

  5. [11]

    Bringmann, Y

    B. Bringmann, Y. Deng, A. R. Nahmod, and H. Yue. Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation. Invent. Math. , 236(3):1133–1411, 2024

    2024

  6. [12]

    Christ, J

    M. Christ, J. Colliander, and T. Tao. Ill-posedness for nonlinear Schr¨ odinger and wave equa- tions. arXiv preprint arXiv:0311048 , 2003

    2003

  7. [14]

    C. Demeter. Fourier restriction, decoupling, and applications . Number 184. Cambridge Uni- versity Press, 2020

    2020

  8. [15]

    Y. Deng. On a bilinear restriction estimate for Schr¨ odinger equations on 2D waveguide. J. Differential Equations, 416:1825–1836, 2025

    2025

  9. [16]

    Y. Deng, B. Di, C. Fan, and Z. Zhao. Bilinear estimate for Schr¨ odinger equation on R × T. arXiv preprint arXiv:2407.05654 , 2024

    2024

  10. [17]

    Y. Deng, C. Fan, K. Yang, Z. Zhao, and J. Zheng. On bilinear Strichartz estimates on waveg- uides with applications. J. Funct. Anal. , 287(9):Paper No. 110595, 2024

    2024

  11. [18]

    Y. Deng, A. Nahmod, and H. Yue. Invariant Gibbs measures and global strong solutions for nonlinear Schr¨ odinger equations in dimension two.Ann. of Math. , 200(2):399–486, 2024

    2024

  12. [19]

    Y. Deng, A. R. Nahmod, and H. Yue. Random tensors, propagation of randomness, and nonlinear dispersive equations. Invent. Math. , 228(2):539–686, 2022. ON RESTRICTED-TYPE STRICHARTZ ESTIMATES 31

    2022

  13. [20]

    Y. Deng, H. Wang, Y. Wang, and Z. Zhao. On long time dynamics of supercritical NLS: Random data theory v.s. Fourier retriction. Preprint, 2025+

    2025

  14. [21]

    B. Dodson. Defocusing nonlinear Schr¨ odinger equations, volume 217. Cambridge University Press, 2019

    2019

  15. [22]

    X. Du, L. Guth, and X. Li. A sharp Schr¨ odinger maximal estimate in R2. Ann. of Math. (2) , 186(2):607–640, 2017

    2017

  16. [23]

    D. Foschi. Inhomogeneous Strichartz estimates. J. Hyperbolic Differ. Equ. , 2(1):1–24, 2005

    2005

  17. [24]

    Ginibre, Y

    J. Ginibre, Y. Tsutsumi, and G. Velo. On the Cauchy problem for the Zakharov system. J. Funct. Anal., 151(2):384–436, 1997

    1997

  18. [25]

    Ginibre and G

    J. Ginibre and G. Velo. Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 133(1):50–68, 1995

    1995

  19. [26]

    Herr and B

    S. Herr and B. Kwak. Strichartz estimates and global well-posedness of the cubic NLS on T2. Forum Math. Pi , 12(e14), 2024

    2024

  20. [27]

    Y. Hong. Strichartz estimates for N -body Schr¨ odinger operators with small potential interac- tions. Discrete Contin. Dyn. Syst. , 37(10):5355–5365, 2017

    2017

  21. [28]

    C. Jao. The energy-critical quantum harmonic oscillator. Comm. Partial Differential Equa- tions, 41(1):79–133, 2016

    2016

  22. [29]

    Keel and T

    M. Keel and T. Tao. Endpoint Strichartz estimates. Amer. J. Math. , 120(5):955–980, 1998

    1998

  23. [30]

    Killip and M

    R. Killip and M. Vi¸ san. Scale invariant Strichartz estimates on tori and applications. Math. Res. Lett., 23(2):445–472, 2016

    2016

  24. [31]

    Killip and M

    R. Killip and M. Vi¸ san. Nonlinear Schr¨ odinger equations at critical regularity. In Evolution equations, volume 17 of Clay Math. Proc. , pages 325–437. American Mathematical Society, Providence, RI, USA, 2013

    2013

  25. [33]

    Kishimoto

    N. Kishimoto. Local well-posedness for the Zakharov system on the multidimensional torus. J. Anal. Math. , 119(1):213–253, 2013

    2013

  26. [34]

    Klainerman and M

    S. Klainerman and M. Machedon. Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. , 46(9):1221–1268, 1993

    1993

  27. [35]

    B. Kwak. Global well-posedness of the energy-critical nonlinear Schr¨ odinger equations on Td. arXiv preprint arXiv:2411.18163 , 2024

    2024 arXiv

  28. [36]

    Kwak and S

    B. Kwak and S. Kwon. Critical local well-posedness of the nonlinear Schr¨ odinger equation on the torus. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 2024

    2024

  29. [37]

    Lindblad and C

    H. Lindblad and C. D. Sogge. On existence and scattering with minimal regularity for semi- linear wave equations. J. Funct. Anal. , 130(2):357–426, 1995

    1995

  30. [38]

    Ozawa and Y

    T. Ozawa and Y. Tsutsumi. Existence and smoothing effect of solutions for the Zakharov equations. Publ. Res. Inst. Math. Sci. , 28(3):329–361, 1992

    1992

  31. [39]

    C. D. Sogge. Fourier integrals in classical analysis , volume 210. Cambridge University Press, 2017

    2017

  32. [40]

    Takaoka and N

    H. Takaoka and N. Tzvetkov. On 2D nonlinear Schr¨ odinger equations with data on R × T. J. Funct. Anal., 182(2):427–442, 2001

    2001

  33. [41]

    T. Tao. Nonlinear dispersive equations: local and global analysis . Number 106. American Mathematical Soc., 2006

    2006

  34. [42]

    Tzvetkov

    N. Tzvetkov. Random data wave equations. Singular random dynamics , 2253:221–313, 2019

    2019

  35. [43]

    Tzvetkov and N

    N. Tzvetkov and N. Visciglia. Small data scattering for the nonlinear Schr¨ odinger equation on product spaces. Comm. Partial Differential Equations , 37(1):125–135, 2012

    2012

  36. [44]

    Y. Wang. Periodic cubic hyperbolic Schr¨ odinger equation onT2. J. Funct. Anal., 265(3):424– 434, 2013

    2013

  37. [45]

    V. E. Zakharov et al. Collapse of Langmuir waves. Sov. Phys. JETP , 35(5):908–914, 1972. 32 YANGKENDI DENG, HAN W ANG, YUZHAO W ANG AND ZEHUA ZHAO Yangkendi Deng Department of Mathematics and Statistics, Beijing Institute of Technology. Key Laboratory of Algebraic Lie Theory a...

    1972

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