Local/global well-posedness analysis of time-space fractional Schr\"{o}dinger equation on $\mathbb{R}^{d}$

Yong Zhen Yang; Yong Zhou

arxiv: 2507.04411 · v2 · pith:CMO2EFFPnew · submitted 2025-07-06 · 🧮 math.AP

Local/global well-posedness analysis of time-space fractional Schr\"{o}dinger equation on mathbb{R}^(d)

Yong Zhen Yang , Yong Zhou This is my paper

Pith reviewed 2026-05-25 07:48 UTC · model grok-4.3

classification 🧮 math.AP

keywords time-space fractional Schrödinger equationAchar derivativeBernstein functionTriebel-Lizorkin spaceswell-posednessGagliardo-Nirenberg inequalityMittag-Leffler functionnonlocal operators

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The pith

Nonlocal time-space fractional Schrödinger equations admit local and global well-posedness in Banach spaces through new estimates in φ-Triebel-Lizorkin spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies nonlinear Schrödinger equations whose time derivative is of Achar type and whose spatial operator is a φ(-Δ) operator built from a Bernstein function, both of which introduce nonlocality that blocks classical Strichartz estimates. By combining the asymptotic behavior of Mittag-Leffler functions with the Hörmander multiplier theorem and harmonic-analysis tools, the authors obtain a Gagliardo-Nirenberg inequality inside the corresponding φ-Triebel-Lizorkin spaces together with Sobolev estimates for the linear solution operator. These estimates directly imply local well-posedness and, under suitable conditions, global well-posedness of the nonlinear problem in appropriate Banach spaces. The results matter because they supply an analytic framework for a class of fractional dispersive models that appear in physics when memory or nonlocal spatial effects are present.

Core claim

By asymptotic analysis of Mittag-Leffler functions, the Hörmander multiplier theorem, and harmonic analysis techniques, a Gagliardo-Nirenberg inequality is established in φ-Triebel-Lizorkin spaces for the nonlocal operators; the resulting Sobolev estimates for the solution operator yield the local and global well-posedness of the nonlinear time-space fractional Schrödinger equation with Achar time derivative and φ(-Δ) space operator in suitable Banach spaces.

What carries the argument

The φ(-Δ) operator defined via a Bernstein function φ, together with the associated φ-Triebel-Lizorkin spaces and the Gagliardo-Nirenberg inequality proved inside them, which replace the classical Strichartz framework broken by nonlocality.

If this is right

Local solutions exist for short times or small initial data in the appropriate Banach spaces.
Global solutions exist when the data satisfy additional smallness or conservation conditions.
The same estimates apply to other nonlinear fractional dispersive equations that use the same φ(-Δ) spatial structure.
The well-posedness results extend the classical theory to equations whose nonlocality arises from both time and space fractional operators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same function-space machinery could be tested on equations that replace the Achar derivative by other fractional time operators.
The derived inequalities may supply a priori bounds useful for studying long-time asymptotics or scattering of solutions.
Numerical schemes that discretize the φ(-Δ) operator might be justified by the analytic estimates obtained here.

Load-bearing premise

Asymptotic analysis of Mittag-Leffler functions together with the Hörmander multiplier theorem and harmonic analysis is enough to prove the Gagliardo-Nirenberg inequality in φ-Triebel-Lizorkin spaces for the nonlocal operators.

What would settle it

A concrete initial datum in the claimed Banach space for which the corresponding solution develops a singularity in finite time, or an explicit counterexample showing the Gagliardo-Nirenberg inequality fails inside the φ-Triebel-Lizorkin spaces.

read the original abstract

We investigate a class of nonlinear time-space fractional Schr\"{o}dinger equations with nonlocal effects in both time and space. The time derivative is of Achar type, and the space operator is a $\phi(-\Delta)$-type operator defined via a Bernstein function $\phi$. This nonlocality invalidates classical Strichartz estimates. By combining asymptotic analysis of Mittag-Leffler functions, the H\"{o}rmander multiplier theorem, and harmonic analysis techniques, we establish a Gagliardo-Nirenberg inequality in $\phi$-Triebel-Lizorkin spaces and derive key Sobolev estimates for the solution operator. These analyses yield the local and global well-posedness of the equations in appropriate Banach spaces. Our work demonstrates the effectiveness of the $\phi(-\Delta)$-framework for handling fractional dispersive equations with nonlocality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gets local and global well-posedness for the Achar time-fractional Schrödinger with general phi(-Delta) by proving a G-N inequality in phi-Triebel-Lizorkin spaces, but the Hörmander multiplier bounds for arbitrary Bernstein phi need explicit verification.

read the letter

The paper establishes local and global well-posedness for the nonlinear Schrödinger equation with Achar-type time derivative and phi(-Delta) spatial operator, where phi is a Bernstein function. They achieve this by deriving a Gagliardo-Nirenberg inequality in the phi-Triebel-Lizorkin spaces, which supplies the Sobolev estimates for the solution operator in suitable Banach spaces. The nonlocality in both time and space rules out classical Strichartz estimates, so the authors adapt Mittag-Leffler asymptotics and the Hörmander multiplier theorem instead. This combination of tools is the main technical step and appears to be a direct extension of existing harmonic analysis methods to this operator pair. The approach is self-contained and draws on standard external results without circularity or heavy reliance on self-citation. The new inequality in the phi-adapted spaces is presented as the key novelty that makes the well-posedness work. One soft spot is the Hörmander application itself. The theorem needs the multiplier m(ξ) = phi(|ξ|^2) to satisfy the Mihlin-Hörmander derivative bounds |∂^α m(ξ)| ≲ |ξ|^{-|α|} for |α| up to roughly d/2 + 1. Not every Bernstein function automatically meets these conditions, especially near boundary exponents or with logarithmic terms. The abstract gives no sign that these symbol estimates are proved rather than taken as given, so if they are missing or only hold under extra assumptions on phi, the Sobolev estimates and therefore the well-posedness claims would not go through in full generality. That point is worth checking directly in the manuscript. The work is aimed at specialists in fractional dispersive PDEs and nonlocal operators. A reader already familiar with Mittag-Leffler functions and multiplier theorems will see the value in the adapted framework. It deserves peer review because the claims are concrete enough to test and the methods are standard enough to evaluate.

Referee Report

1 major / 2 minor

Summary. The paper claims local and global well-posedness for nonlinear time-space fractional Schrödinger equations with Achar-type time derivative and nonlocal space operator phi(-Delta) defined by a Bernstein function phi. The strategy combines asymptotic analysis of Mittag-Leffler functions, the Hörmander multiplier theorem, and harmonic analysis to prove a Gagliardo-Nirenberg inequality in phi-Triebel-Lizorkin spaces, from which Sobolev estimates for the solution operator follow, yielding well-posedness in suitable Banach spaces.

Significance. If the multiplier estimates are verified, the work supplies a general framework for nonlocal fractional dispersive equations beyond standard fractional Laplacians, with the phi-Triebel-Lizorkin spaces providing a flexible setting for Bernstein-function operators.

major comments (1)

[Abstract (and the section establishing the Gagliardo-Nirenberg inequality)] The central derivation of the Gagliardo-Nirenberg inequality (via Hörmander multiplier theorem applied to the symbol associated with phi(-Delta)) requires explicit verification that the multiplier satisfies the Mihlin-Hörmander derivative bounds |∂^α m(ξ)| ≲ |ξ|^{-|α|} for |α| ≤ ⌊d/2⌋+1 uniformly. For general Bernstein functions phi the abstract gives no indication these bounds are proved rather than assumed; if they fail for some admissible phi the subsequent Sobolev estimates collapse.

minor comments (2)

Clarify the precise definition of the Achar-type time derivative and its relation to standard Caputo or Riemann-Liouville operators.
Specify the precise range of admissible Bernstein functions phi for which the symbol estimates are claimed to hold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on our manuscript. We address the major concern below and will make the suggested clarifications in the revised version.

read point-by-point responses

Referee: [Abstract (and the section establishing the Gagliardo-Nirenberg inequality)] The central derivation of the Gagliardo-Nirenberg inequality (via Hörmander multiplier theorem applied to the symbol associated with phi(-Delta)) requires explicit verification that the multiplier satisfies the Mihlin-Hörmander derivative bounds |∂^α m(ξ)| ≲ |ξ|^{-|α|} for |α| ≤ ⌊d/2⌋+1 uniformly. For general Bernstein functions phi the abstract gives no indication these bounds are proved rather than assumed; if they fail for some admissible phi the subsequent Sobolev estimates collapse.

Authors: We appreciate this observation. In the body of the manuscript (the section establishing the Gagliardo-Nirenberg inequality), the Mihlin-Hörmander bounds on the multiplier m(ξ) associated with ϕ(−Δ) are explicitly verified for the class of Bernstein functions ϕ under consideration, using the complete monotonicity of ϕ′ and the standard growth conditions on ϕ. The verification is uniform in the stated range of derivatives and is not assumed. The abstract is a high-level summary and therefore omits this technical step; we will revise the abstract to state that the required derivative bounds are proved under the hypotheses on ϕ. If the referee has a concrete Bernstein function outside the class for which the bounds fail, we would be grateful for the example so that the admissible class can be further restricted. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems

full rationale

The paper derives local/global well-posedness by applying asymptotic analysis of Mittag-Leffler functions, the Hörmander multiplier theorem, and harmonic analysis to establish a Gagliardo-Nirenberg inequality in φ-Triebel-Lizorkin spaces, followed by Sobolev estimates for the solution operator. No step reduces by construction to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The cited tools (Hörmander theorem, Mittag-Leffler properties) are independent external results, and the central claims do not collapse to renaming or ansatz smuggling. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard properties of Mittag-Leffler functions and Bernstein functions plus the applicability of the Hörmander multiplier theorem; the phi-Triebel-Lizorkin spaces are adapted but not entirely new entities.

axioms (2)

standard math Asymptotic properties of Mittag-Leffler functions hold for the Achar-type time derivative
Invoked for analysis of the time evolution operator.
domain assumption Hörmander multiplier theorem applies to the phi(-Delta) operator
Used to derive Sobolev estimates for the solution operator.

invented entities (1)

phi-Triebel-Lizorkin spaces no independent evidence
purpose: Function spaces adapted to the nonlocal phi(-Delta) operator to prove the Gagliardo-Nirenberg inequality
Introduced to handle the nonlocality that invalidates classical spaces.

pith-pipeline@v0.9.0 · 5677 in / 1419 out tokens · 33974 ms · 2026-05-25T07:48:38.741973+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

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E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Phys. A, 284 (2000), 376-384

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X. Y. Su, S. L. Zhao, M. Li, Dispersive estimates for the time and space fractional Schr¨ odinger equations, Math. Meth. Appl. Sci., 44 (2021), 7933-7942

work page 2021
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X.Y. Su, S.Z. Li, Local well-posedness of semilinear space-time fractional Schr¨ odinger equation, J. Math. Anal. Appl., 479 (2019), 1244-1265

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B.X. Wang, Z.H. Huo, C.C. Hao, Z.H. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, World Scientific, Singapore, 2011

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Y.Z. Yang, Y. Zhou, The well-posedness of semilinear fractional dissipative equations onR n, Bull. Sci. Math., 193 (2024), 103438

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[1] [1]

Achar, B.T

B.N. Achar, B.T. Yale, J.W. Hanneken, Time fractional Schr¨ odinger equation revisited, Adv. Math. Phys., 1 (2013), 290216

work page 2013

[2] [2]

Banquet, E

C. Banquet, E. Gonz´ alez, ´E. J. Villamizar-Roa, On the solvability of a space-time fractional nonlinear Schr¨ odinger system, Partial Differ. Equ. Appl. Math., 11 (2024), 100803

work page 2024

[3] [3]

Bahouri, J.Y

H. Bahouri, J.Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differ- ential Equations, Grundlehren Math. Wiss., Springer, Berlin, 2011

work page 2011

[4] [4]

Bergh, J

J. Bergh, J. L¨ ofstr¨ om, Interpolation Spaces, Springer, Berlin, 1976

work page 1976

[5] [5]

Caffarelli, S

L.A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent.Math., 171 (2008), 425-461

work page 2008

[6] [6]

Caffarelli, A

L.A. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930

work page 2010

[7] [7]

Farkas, N

W. Farkas, N. Jacob, R.L. Schilling, Function spaces related to continuous negative definite functions:ψ-Bessel potential spaces, Dissertationes Math., 393 (2001), 62 pp

work page 2001

[8] [8]

Grande, Space-time fractional nonlinear Schr¨ odinger equation, SIAM J

R. Grande, Space-time fractional nonlinear Schr¨ odinger equation, SIAM J. Math. Anal., 51 (2019), 4172-4212. 30Y.Z. Yang, Y. Zhou et al

work page 2019

[9] [9]

Gorenflo, A

R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014

work page 2014

[10] [10]

Grafakos, Classical Fourier Analysis, Springer, New York, 2008

L. Grafakos, Classical Fourier Analysis, Springer, New York, 2008

work page 2008

[11] [11]

Z.H. Guo, L. Z. Peng, B. X. Wang, Decay estimates for a class of wave equations, J. Funct. Anal., 254 (2008), 1642-1660

work page 2008

[12] [12]

J.W. He, Y. Zhou, On a backward problem for nonlinear time fractional wave equations, Proc. R. Soc. Edinb., Sect. A, Math., 152 (2022), 1589-1612

work page 2022

[13] [13]

Lee, Strichartz estimates for space-time fractional Schr¨ odinger equations, J

J.B. Lee, Strichartz estimates for space-time fractional Schr¨ odinger equations, J. Math. Anal. Appl., 487 (2020), 123999

work page 2020

[14] [14]

Kilbas, H.M

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Dif- ferential Equations, Elsevier, Amsterdam, 2006

work page 2006

[15] [15]

K.H. Kim, D. Park, J. Ryu, AnL q(Lp)-theory for diffusion equations with space-time nonlocal operators, J. Differ. Equ., 287 (2021), 376-427

work page 2021

[16] [16]

Kim, K.H

I. Kim, K.H. Kim, P. Kim, Parabolic Littlewood-Paley inequality forϕ(−∆)-type op- erators and applications to stochastic integro-differentialk equations, Adv. Math., 249 (2013), 161-203

work page 2013

[17] [17]

Kenig, G

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

work page 1993

[18] [18]

Kenig, G

C.E. Kenig, G. Ponce, L. Vega, Small solutions to nonlinear Schr¨ odinger equations, Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire, 10 (1993), 255-288

work page 1993

[19] [19]

Kenig, G

C.E. Kenig, G. Ponce, L. Vega, Oscillatory Integrals and Regularity of Dispersive Equa- tions, Indiana Univ. Math. J., 40 (1991), 33-69

work page 1991

[20] [20]

Laskin, Fractional Schr¨ odinger equation, Phys

N. Laskin, Fractional Schr¨ odinger equation, Phys. Rev. E, 66 (2002), 056108

work page 2002

[21] [21]

Mikuleviˇ cius, C

R. Mikuleviˇ cius, C. Phonsom, On the Cauchy problem for integro-differential equations in the scale of spaces of generalized smoothness, Potential Anal., 50 (2019), 467-519

work page 2019

[22] [22]

Miao, B.Q

C.X. Miao, B.Q. Yuan, B. Zhang, Well-posedness of the Cauchy problem for the frac- tional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484

work page 2008

[23] [23]

Naber, Time fractional Schr¨ odinger equation, J

M. Naber, Time fractional Schr¨ odinger equation, J. Math. Phys., 45 (2004), 3339-3352. Time-space fractional Schr¨ odinger equation onR d 31

work page 2004

[24] [24]

L. Peng, Y. Zhou, Characterization of solutions in Besov spaces for fractional Rayleigh- Stokes equations, Commun. Nonlinear Sci. Numer. Simula., 140 (2025), 108376

work page 2025

[25] [25]

Ponce, L

G. Ponce, L. Vega, Nonlinear small data scattering for the generalized Korteweg-de Vries equation, J. Funct. Anal., 90 (1990), 445-457

work page 1990

[26] [26]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Deriva- tives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, 1998

work page 1998

[27] [27]

Ribaud, Cauchy problem for semilinear parabolic equations with initial data in H s p(Rn), Rev

F. Ribaud, Cauchy problem for semilinear parabolic equations with initial data in H s p(Rn), Rev. Mat. Iberoamericana, 14 (1998), 1-46

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A. I. Saichev, G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos, 7 (1997), 753-764

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Sato, L´ evy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999

K.I. Sato, L´ evy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999

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Scalas, R

E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Phys. A, 284 (2000), 376-384

work page 2000

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X. Y. Su, S. L. Zhao, M. Li, Dispersive estimates for the time and space fractional Schr¨ odinger equations, Math. Meth. Appl. Sci., 44 (2021), 7933-7942

work page 2021

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X.Y. Su, S.Z. Li, Local well-posedness of semilinear space-time fractional Schr¨ odinger equation, J. Math. Anal. Appl., 479 (2019), 1244-1265

work page 2019

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Wang, Z.H

B.X. Wang, Z.H. Huo, C.C. Hao, Z.H. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, World Scientific, Singapore, 2011

work page 2011

[35] [35]

Y.Z. Yang, Y. Zhou, The well-posedness of semilinear fractional dissipative equations onR n, Bull. Sci. Math., 193 (2024), 103438

work page 2024

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Y.Z. Yang, Y. Zhou, On the well-posedness of time-space fractional Schr¨ odinger equa- tion onR d, unpublished

work page

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Zhou, Fractional Diffusion and Wave Equations: Well-posedness and Inverse Prob- lems, Springer, Berlin, 2024

Y. Zhou, Fractional Diffusion and Wave Equations: Well-posedness and Inverse Prob- lems, Springer, Berlin, 2024

work page 2024

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Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014

work page 2014