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Critical scattering for the nonlinear Schr\"odinger equation on waveguide manifolds

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Pith's one-line read This paper proves small-data global scattering for the nonlinear Schrödinger equation on waveguide manifolds $\mathbb{R}^m \times \mathbb{T}^n$ for every mass-supercritical power $\alpha > 4/m$, working in the critical Sobolev space…

desk verdict Proves a major waveguide scattering theorem, but the proof leans on two central estimates imported from a to-appear paper and a 'straightforward' lemma with omitted details; verify those before trusting the theorem. read the letter →

arxiv 2506.00442 v2 pith:OG6LW3KP submitted 2025-05-31 math.AP math.CA

Yongming Luo This is my paper
classification math.APmath.CA MSC 35Q5535B4035B33
keywords nonlinearSchrödingerequationsmalldatascatteringwaveguidemanifoldcriticalSobolevspaceBesovspacesStrichartzestimatesmass-supercriticalnonlinearityatomic
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 establishes small-data scattering for the nonlinear Schrödinger equation (NLS) on waveguide manifolds $\mathbb{R}^m \times \mathbb{T}^n$: for any nonlinearity power $\alpha > 4/m$, sufficiently small initial data in the critical Sobolev space $H^{s_c}$ produce a global solution that asymptotically becomes a free wave. This settles the critical small-data scattering problem for arbitrary mass-supercritical powers, including non-algebraic nonlinearities, on all waveguides outside the mass-critical endpoint. The result matters because waveguide manifolds mix dispersive Euclidean directions with confining periodic directions, and prior critical-space methods either required algebraic nonlinearities or failed at high frequencies. The proof supplies a new anisotropic Besov-space framework with fixed-point arguments, where earlier approaches only worked for algebraic nonlinearities or in the subcritical regime.

What carries the argument

The central object is the critical space $Z^s$ (with $s = s_c$), built from frequency-localized Strichartz norms with Besov regularity in time and in the periodic and Euclidean directions, together with the atomic space $Y^s$. The argument is carried by two estimates: the trilinear bound $\|vw|u|^{\alpha-k}u^k\|_{L^1} \lesssim \|v\|_{Z^0}\|w\|_{Z^0}\|u\|_{Z^s}^{\alpha}$ (Lemma 5.5) and the dual nonlinear bound $\||u|^\alpha u\|_{(Z^{-s})'} \lesssim \|u\|_{Z^s}^{1+\alpha}$ (Lemma 6.3). These rest on anisotropic Strichartz estimates with almost unlimited exponents from [27], anisotropic fractional chain rules in vector-valued Besov spaces (Lemmas 3.7, 3.13), and a fixed-point argument in a complete metric space whose metric is the weaker $Y^0$ norm.

What would settle it

Check Lemma 6.3 directly: for a single high-frequency mode $u$ oscillating only in the periodic directions, compute $\||u|^\alpha u\|_{(Z^{-s})'}$ and $\|u\|_{Z^s}^{1+\alpha}$; if the ratio is unbounded as the frequency grows, the theorem fails in that dimension. More broadly, a counterexample to the dyadic summation in [18, Lem. 4.4] within the $Z^s$ norm would invalidate the proof.

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Extended reading notes

Core claim

Theorem 1.1 states: for $\alpha > 4/m$, $d = m+n$, $s_c = d/2 - 2/\alpha$ and $s_c < 1+\alpha$, there exists $\delta > 0$ such that every $u_0 \in H^{s_c}(\mathbb{R}^m \times \mathbb{T}^n)$ with $\|u_0\|_{H^{s_c}} \le \delta$ admits a global solution $u \in C(\mathbb{R}; H^{s_c})$ of $(i\partial_t + \Delta_{x,y})u = \pm|u|^\alpha u$, and this solution scatters: there are $\phi_\pm \in H^{s_c}$ with $\|u(t) - e^{it\Delta}\phi_\pm\|_{H^{s_c}} \to 0$ as $t \to \pm\infty$. The discovery is that the small-data critical problem is solvable for the whole mass-supercritical range, $\alpha > 4/m$, on every waveguide manifold, with a proof that treats the non-algebraic factor $|u|^\alpha$ as an independent object and gains the necessary cancellation from Besov regularity in time instead of from frequency-space algebra.

Load-bearing premise

The proof imports, without reproducing it, the key dual-norm bound for the nonlinearity from the periodic setting, and the whole contraction argument falls apart if that bound does not transfer to the waveguide critical norm.

Editorial extensions

If this is right

  • For every waveguide manifold $\mathbb{R}^m \times \mathbb{T}^n$ and every $\alpha > 4/m$, the critical NLS is small-data globally well-posed and scattering in $H^{s_c}$, filling the mass-supercritical range left open by earlier product-space results.
  • The fixed-point construction gives uniqueness of the scattering solution inside the space $K$, not just existence.
  • The framework extends the periodic-NLS technology of [18] to anisotropic and vector-valued settings, so the two-object treatment of $|u|^\alpha$ and $u$ can be reused elsewhere.
  • The mass-critical endpoint $\alpha = 4/m$ remains open here, but the paper notes anisotropic endpoint Strichartz estimates do hold, so the method pinpoints a concrete next target.
  • The author suggests that combining this small-data machinery with concentration-compactness ideas could yield large-data energy-critical scattering on waveguides.

Reading between the lines

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

  • If the quoted periodic nonlinear estimate genuinely transfers to the waveguide $Z^s$ norm, the same architecture likely works for other semilinear dispersive models on product manifolds, such as Hartree or derivative NLS, where the nonlinearity is not algebraic.
  • The mass-critical endpoint conjecture could be tested by trying the same fixed-point scheme with endpoint-admissible anisotropic Strichartz estimates; failure would indicate a genuine cancellation deficit rather than a technical one.
  • The explicit condition $s_c < 1+\alpha$ shows the regularity ceiling is set by the nonlinearity's Hölder regularity, a constraint that should appear in any vector-valued Besov approach to non-algebraic NLS.
  • The completeness of $K$ under the $Y^0$ metric might be exported to prove uniqueness in larger natural classes for related equations.
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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

4 major / 5 minor

Summary. The paper claims a small-data scattering theorem for the nonlinear Schr\"odinger equation with power nonlinearity |u|^\alpha u on waveguide manifolds R^m \times T^n. The main result, Theorem 1.1, asserts global scattering in H^{s_c} for \alpha > 4/m and s_c < 1 + \alpha, provided the initial data have sufficiently small H^{s_c} norm. The proof introduces an anisotropic Besov-atomic space Z^s, develops multilinear estimates for the dual of the nonlinearity, and closes a fixed-point argument in a Y^s/Z^s ball. Several auxiliary tools are provided, including vector-valued fractional chain and product rules, anisotropic Strichartz estimates, and a Galilean-invariance argument. The paper does not cover the mass-critical endpoint \alpha = 4/m, which is explicitly acknowledged in Remark 1.2.

Significance. If the central estimates are fully justified, the result is a substantial extension of the small-data scattering theory for NLS on waveguides: it covers arbitrary mass-supercritical powers, including non-algebraic nonlinearities, and it does so without the endpoint restrictions of the Hani-Pausader Strichartz estimates. The fixed-point formulation in Section 7 is a genuine alternative to the approximation methods of [18], and the anisotropic Z^s framework is well adapted to the product-space geometry. The main caveat is that two load-bearing nonlinear estimates are imported without proof from the to-appear paper [18], and one key Besov chain rule is stated with details omitted. These gaps are correctness risks, not merely presentational issues, and they should be addressed before the paper can be accepted as a standalone contribution.

major comments (4)
  1. [Section 6, Lemma 6.3] The central estimate \| |u|^\alpha u \|_{(Z^{-s})'} \lesssim \|u\|_{Z^s}^{1+\alpha} is stated in Lemma 6.3 as an immediate consequence of Lemma 6.1 and Lemma 6.2, with the proof deferred to [18, Lem. 4.4]. This is not an innocuous reference: the norm in (4.1) has anisotropic \ell^{s,2} weights over Euclidean dyadic scales, separate torus scales, and Besov-in-time factors, so the combinatorial summation over K and N used on the torus must be re-verified for this norm. Since Lemma 6.3 is used in Section 7 both to prove that the Duhamel map sends K into itself and to prove the scattering limit (1.2), the main theorem is not self-contained at its decisive step.
  2. [Section 6, Lemma 6.1] Lemma 6.1 is likewise imported from [18, Lem. 4.1] without proof. The estimate for F^N_K in the low-high regime K \lesssim N does not follow formally from Corollary 5.6 alone, because one must sum over N and compare the frequency-localized Z^{-s} dual norm with the global Z^s norm of u; the text gives no argument for this comparison in the waveguide setting. A proof, or at least a precise transfer theorem from the periodic case, is needed.
  3. [Section 3.2, Lemma 3.7] Lemma 3.7 is introduced as a vector-valued generalization of [18, Lem. 2.8] with the comment that the modification is straightforward and details are omitted. This lemma is used in the proof of Lemma 5.5 to control regime III, and it is precisely the kind of vector-valued Besov chain rule that the paper identifies as the main obstacle in the anisotropic setting. The proof of (3.2) should be written out explicitly.
  4. [Section 5.3, Lemma 5.5 proof] The exponent computations in the estimates of the terms III and IV are not fully checkable. In the paragraph following (5.8), the number r_{tx,1} is introduced implicitly, and in (5.9) the parameter \theta is used before it is defined two lines later. Because the admissibility identities in this paragraph determine the final \ell^{s,2} weights, the complete computation should be displayed.
minor comments (5)
  1. [Section 2, Littlewood-Paley projectors] The support conditions for \eta_1 are inconsistent as written: \eta(t) \equiv 1 for |t| \le 1 and \eta(t) \equiv 0 for |t| \ge 0 cannot both hold. The second condition should be |t| \ge 2 or a similar disjoint region.
  2. [Lemma 3.9 and Lemma 3.13] The symbol 'm' is used both for the Euclidean dimension and for an integer parameter in the statements of these lemmas. Please rename the integer parameter to avoid ambiguity.
  3. [Proof of Lemma 5.5] The notation r_{2y} and r_{y,2} appears to refer to the same quantity; please unify these symbols.
  4. [Section 4, definition of Z^s] In (4.1), the notation 'max_{p \in \max\{m^*, 1/\sigma_1\}}' does not specify a range for p. It should be written as a supremum or maximum over p \ge \max\{m^*, 1/\sigma_1\}.
  5. [References] Reference [17] is cited without publication data; please update the citation if a journal version or arXiv number is available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fixed-point and scattering argument rests on independent multilinear estimates; the main gap is deferred proofs from Kwak-Kwon [18], which is a completeness/reproducibility issue, not a circular reduction.

full rationale

The paper's central claim is Theorem 1.1, proved by a contraction-mapping argument in the complete metric space K defined in (7.1). The contraction estimate uses Lemma 5.5 and the scattering limit uses Lemma 6.3, but neither reduces to the theorem's conclusion by definition. The Z^s norm in (4.1) is a linear function-space norm built from Strichartz and Besov factors; it is not defined in terms of the nonlinearity |u|^alpha u or in terms of the scattering statement, so Lemma 6.3, asserting the dual-space bound for the nonlinearity, is a substantive estimate rather than a tautology. The key external debts are to Kwak-Kwon [18]: Lemma 6.1 states 'For a proof, see [18, Lem. 4.1]', Lemma 6.3 states 'For a proof, see [18, Lem. 4.4]', and Lemma 3.7 states 'The modification is straightforward, we omit the details.' These are omitted proofs and potential correctness risks because the waveguide Z^s norm is anisotropic, but [18] is prior work by different authors and the deferred estimates are not parameters fitted to the target result. The author's own prior works [19]-[22] appear only in contextual remarks, notably Remark 1.5 about expected large-data results, and do not close the fixed-point argument. No prediction is a fitted input, no uniqueness theorem is imported from the author's own prior work, and no known result is merely renamed. Accordingly, the derivation chain is not circular; the appropriate finding is no significant circularity, even though the paper is not fully self-contained with respect to [18].

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central claim rests on external harmonic-analysis theorems and on unproved-in-this-text adaptations of Kwak-Kwon lemmas. The only hand-chosen parameters are smallness constants sigma and sigma_1, which are not fitted to data and do not affect the statement of the theorem. No new physical entities are introduced; the function spaces X^s, Y^s, Z^s are analytic constructions, not empirically testable objects.

free parameters (1)
  • sigma and sigma_1 = sufficiently small, 0 < sigma << sigma_1 << 1
    Smallness parameters in the Z^s norm, Besov exponents, admissible Strichartz pairs, and the interpolation arguments. The proof only requires existence of such sufficiently small positive constants, not numerical values.
assumptions (7)
  • standard math Tzvetkov-Visciglia Strichartz estimates (3.8) and (3.9) for the linear Schrodinger flow on R^m x T^n hold for the stated admissible pairs.
    Cited from [27] and used throughout Sections 3.3 and 4. This is the anisotropic replacement for periodic Strichartz estimates and underlies the Z^s norm.
  • standard math Atomic U^2 and V^2 embeddings, duality, and the transfer principle of Hadac-Herr-Koch and Herr-Tataru-Tzvetkov.
    Used in Lemmas 2.1, 2.2, and 3.15 to define X^s, Y^s, N^s and to transfer Strichartz estimates to those spaces.
  • standard math Besov space difference characterizations, interpolation, duality, and operator-valued Mikhlin multiplier theorems.
    Used in Lemmas 3.1 to 3.4 and 3.12, and extensively in Sections 5 and 6. Cited to [1,2,23,24,26].
  • standard math Fractional chain and product rules of Kenig-Ponce-Vega for the Euclidean directions.
    Lemmas 3.10 and 3.11 are stated as [16] and are the basis for the new anisotropic Besov chain rule in Lemma 3.13.
  • standard math The Kwak-Kwon lemmas, including the frequency-combination estimates stated here as Lemma 6.1 and Lemma 6.3, are correct and remain valid after translation to the waveguide setting.
    The paper imports these central nonlinear estimates from [18] with 'for a proof, see [18]' and does not reproduce the combinatorial arguments. This is the main external dependency.
  • domain assumption Smallness in H^{s_c} of the initial data implies the linear evolution lies in Y^{s_c} with comparable norm.
    Used at the start of the proof of Theorem 1.1. The equality of the V^2 norm of e^{itΔ}u0 with ||u0||_{L^2} after frequency-cube localization is standard but not shown.
  • domain assumption The model is the power nonlinearity |u|^alpha u on R^m x T^n with alpha > 4/m and sc < 1 + alpha.
    These are the hypotheses of Theorem 1.1. The mass-critical endpoint alpha = 4/m is explicitly excluded.

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Pith. "Pith review of Critical scattering for the nonlinear Schr\"odinger equation on waveguide manifolds." pith.science (2026). https://pith.science/paper/OG6LW3KP

@misc{pith2026250600442,
  author       = {Pith},
  title        = {Pith review of: Critical scattering for the nonlinear Schr\"odinger equation on waveguide manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OG6LW3KP}},
  note         = {Machine review of arXiv:2506.00442}
}
read the original abstract

We study the small data scattering problem in critical spaces for the nonlinear Schr\"odinger equation (NLS) on waveguide manifolds. Our work is primarily inspired by the recent paper of Kwak and Kwon \cite{KwakKwon} that established the local well-posedness of the periodic NLS with possibly non-algebraic nonlinearity. While we adopt a framework similar to \cite{KwakKwon} for our problem, two main obstacles prevent its direct adaptation to the waveguide setting. First, the classical Strichartz estimates for NLS in critical product spaces, introduced by Hani and Pausader, possess limited endpoints and are thus inapplicable to high-dimensional waveguides. Second, the crucial fractional arguments used in \cite{KwakKwon} rely on a well-known fractional derivative formula due to Strichartz, which admits only a Hilbert space-valued extension and is therefore incompatible with our model setting. To overcome these difficulties, we develop an anisotropic generalization of the framework in \cite{KwakKwon} using the anisotropic Strichartz estimates established by Tzvetkov and Visciglia, which allow for nearly unlimited endpoints. We also resolve several new challenges arising from the vector-valued and anisotropic nature of the model by employing novel interpolation techniques within Besov spaces. As a further novelty, we provide a new proof of the main result based on classical fixed point arguments, differing from the approximation methods used in \cite{KwakKwon}. Consequently, we settle the small data scattering problem in critical spaces for the NLS with arbitrary mass-supercritical nonlinearity on waveguide manifolds.

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Cited by 1 Pith paper

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

  1. Almost sure local well-posedness for the nonlinear Schr\"odinger equations on $\Bbb T^d$ with non-algebraic nonlinearity

    math.AP 2026-08 conditional novelty 8.0 of 10

    Random initial data below the critical Sobolev regularity still support local solutions to the periodic nonlinear Schrodinger equation for any non-algebraic power in the mass-supercritical range.

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Works this paper leans on

32 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [18]

    Kwak, B., and Kwon, S.Critical local well-posedness of the nonlinear Schr¨ odinger equation on the torus. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire(to appear). CRITICAL SCATTERING FOR THE NLS ON W A VEGUIDE MANIFOLD 24

    work page

  2. [1]

    Amann, H.Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186 (1997), 5–56

    work page 1997

  3. [2]

    Compact embeddings of vector-valued Sobolev and Besov spaces

    Amann, H. Compact embeddings of vector-valued Sobolev and Besov spaces. Glas. Mat. Ser. III 35(55), 1 (2000), 161–177. Dedicated to the memory of Branko Najman

    work page 2000

  4. [3]

    PDE 14 , 4 (2021), 1125–1152

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

    work page 2021

  5. [4]

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

    work page 1993

  6. [5]

    Chen, X., and Holmer, J.The derivation of the T3 energy-critical NLS from quantum many-body dynamics. Invent. Math. 217 , 2 (2019), 433–547

    work page 2019

  7. [6]

    Forum Math

    Chen, X., and Holmer, J.Unconditional uniqueness for the energy-critical nonlinear Schr¨ odinger equation on T4. Forum Math. Pi 10 (2022), 49. Id/No e3

    work page 2022

  8. [7]

    Chen, X., Shen, S., and Zhang, Z.The unconditional uniqueness for the energy-supercritical NLS. Ann. PDE 8 , 2 (2022), 82. Id/No 14

    work page 2022

Show all 32 references
  1. [8]

    Cheng, X., Guo, Z., Yang, K., and Zhao, L.On scattering for the cubic defocusing nonlinear Schr¨ odinger equation on the waveguideR2 × T. Rev. Mat. Iberoam. 36 , 4 (2020), 985–1011

    2020

  2. [9]

    Cheng, X., Guo, Z., and Zhao, Z.On scattering for the defocusing quintic nonlinear Schr¨ odinger equation on the two-dimensional cylinder. SIAM J. Math. Anal. 52 , 5 (2020), 4185–4237

    2020

  3. [10]

    Colliander, J., Keel, M., Staffilani, G., Takaoka, H., and Tao, T.Global well-posedness and scattering for the energy-critical nonlinear Schr¨ odinger equation inR3. Ann. of Math. (2) 167 , 3 (2008), 767–865

    2008

  4. [11]

    Colliander, J., Keel, M., Staffilani, G., Takaoka, H., and Tao, T.Transfer of energy to high frequencies in the cubic defocusing nonlinear Schr¨ odinger equation.Invent. Math. 181 , 1 (2010), 39–113

    2010

  5. [12]

    Hadac, M., Herr, S., and Koch, H.Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 26, 3 (2009), 917–941

    2009

  6. [13]

    Hani, Z., and Pausader, B.On scattering for the quintic defocusing nonlinear Schr¨ odinger equation on R × T2. Comm. Pure Appl. Math. 67 , 9 (2014), 1466–1542

    2014

  7. [14]

    Forum Math

    Herr, S., and Kwak, B.Strichartz estimates and global well-posedness of the cubic NLS on T2. Forum Math. Pi 12 (2024), Paper No. e14, 21

    2024

  8. [15]

    Duke Math

    Herr, S., Tataru, D., and Tzvetkov, N.Global well-posedness of the energy-critical nonlinear Schr¨ odinger equation with small initial data inH 1(T3). Duke Math. J. 159 , 2 (2011), 329–349

    2011

  9. [16]

    E., Ponce, G., and Vega, L.Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle

    Kenig, C. E., Ponce, G., and Vega, L.Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46 , 4 (1993), 527–620

    1993

  10. [17]

    Kwak, B.Global well-posedness of the energy-critical nonlinear Schr¨ odinger equations onTd, 2024

    2024

  11. [19]

    Luo, Y.Normalized ground states and threshold scattering for focusing NLS on Rd × T via semivirial- free geometry, 2022

    2022

  12. [20]

    Luo, Y.On long time behavior of the focusing energy-critical NLS on Rd × T via semivirial-vanishing geometry. J. Math. Pures Appl. (9) 177 (2023), 415–454

    2023

  13. [21]

    Sharp scattering for focusing intercritical NLS on high-dimensional waveguide manifolds

    Luo, Y. Sharp scattering for focusing intercritical NLS on high-dimensional waveguide manifolds. Math. Ann. (May 2023)

    2023

  14. [22]

    Almost sure scattering for the defocusing cubic nonlinear Schr¨ odinger equation on R3 × T

    Luo, Y. Almost sure scattering for the defocusing cubic nonlinear Schr¨ odinger equation on R3 × T. J. Funct. Anal. 287 , 4 (2024), Paper No. 110492, 33

    2024

  15. [23]

    R.On Fourier multiplier transformations of Banach-valued functions

    McConnell, T. R.On Fourier multiplier transformations of Banach-valued functions. Trans. Amer. Math. Soc. 285 , 2 (1984), 739–757

    1984

  16. [24]

    130 (2016), 138–156

    Nakamura, M., and W ada, T.Modified Strichartz estimates with an application to the critical nonlinear Schr¨ odinger equation.Nonlinear Anal. 130 (2016), 138–156

    2016

  17. [25]

    S.Multipliers on fractional Sobolev spaces

    Strichartz, R. S.Multipliers on fractional Sobolev spaces. J. Math. Mech. 16 (1967), 1031–1060

    1967

  18. [26]

    Theory of function spaces , vol

    Triebel, H. Theory of function spaces , vol. 78 of Monographs in Mathematics . Birkh¨ auser Verlag, Basel, 1983

    1983

  19. [27]

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

    2012

  20. [28]

    Tzvetkov, N., and Visciglia, N.Well-posedness and scattering for nonlinear Schr¨ odinger equations on Rd × T in the energy space. Rev. Mat. Iberoam. 32 , 4 (2016), 1163–1188

    2016

  21. [29]

    Visan, M.The defocusing energy-critical nonlinear Schr¨ odinger equation in higher dimensions.Duke Math. J. 138 , 2 (2007), 281–374

    2007

  22. [30]

    PhD thesis, Uni- versit¨ at Z¨ urich, 2003

    W alker, C.On Diffusive and Non-Diffusive Coalescence and Breakage Processes . PhD thesis, Uni- versit¨ at Z¨ urich, 2003

    2003

  23. [31]

    Zhao, Z.On scattering for the defocusing nonlinear Schr¨ odinger equation on waveguideRm ×T (when m = 2, 3). J. Differential Equations 275 (2021), 598–637

    2021

  24. [32]

    Zhao, Z., and Zheng, J.Long time dynamics for defocusing cubic nonlinear Schr¨ odinger equations on three dimensional product space. SIAM J. Math. Anal. 53 , 3 (2021), 3644–3660. Yongming Luo F aculty of Computational Mathematics and Cybernetics Shenzhen MSU-BIT University, Ch...

    2021

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