Probabilistic global-wellposedness for the energy-supercritical Schr\"odinger equations on compact manifolds

Filone G. Longmou-Moffo; Mouhamadou Sy; Seynabou Gueye

REVIEW 3 major objections 5 minor 64 references

Probabilistic global-wellposedness for the energy-supercritical Schr\"odinger equations on compact manifolds

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For every compact Riemannian manifold of dimension at least 3, the energy-supercritical nonlinear Schrödinger equation has almost-sure global solutions with invariant measures and slow Sobolev-norm growth, even for singular data of…

desk verdict Serious IID-limit extension to compact manifolds with a real, repairable gap in the full-measure ensemble lemma. read the letter →

arxiv 2502.08812 v1 pith:S6QKSZ53 submitted 2025-02-12 math.AP

Seynabou Gueye , Filone G. Longmou-Moffo , Mouhamadou Sy This is my paper · ORCID

classification math.AP MSC 35A0135Q5535R1160H1537K0637L50

keywords nonlinearSchrödingerequationenergysupercriticalglobalwell-posednessinvariantmeasurecompactRiemannianmanifoldsingularSobolevspacesinviscid-infinite-dimensionallimitstatisticalensemble

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 aims to prove that the energy-supercritical nonlinear Schrödinger equation on a general compact Riemannian manifold of dimension at least 3 admits a probabilistic global theory: for almost every initial datum drawn from a specially constructed invariant measure, the solution exists for all time even though the datum may be too rough for classical energy-based well-posedness. This would bypass the standard obstruction that supercritical nonlinearities cannot be controlled by the linear evolution, replacing a conservation law with an invariant measure as the globalizing mechanism. The main theorem asserts the existence, for every admissible singular Sobolev order $s\in(s_{M^d},d/2]$ and every sufficiently large power $q$, of a full-measure set $\Sigma$ on which the NLS flow is global, maps $\Sigma$ to itself, grows at most like $(1+|t|)^\varepsilon$ in the $H^{s-}$ norm, and leaves the measure invariant, with the $L^2$-norm distribution absolutely continuous. The proof follows an inviscid-infinite-dimensional-limit strategy: build invariant measures for damped stochastic Galerkin approximations, pass to the inviscid limit, and select a statistically typical ensemble of initial data with uniform a priori bounds.

What carries the argument

Three devices carry the argument. First, a local well-posedness theory for the Galerkin-truncated equation, based on Strichartz estimates that lose a fraction of a derivative on general compact manifolds but not on tori or Zoll manifolds; this sets the regularity thresholds $s_{M^d}$. Second, a dissipation operator $L_s(u)=(-\Delta)^{s-1}u+C_{d,s}\|u\|_{H^{s-}}^{3\tilde{k}-1}u$ inserted into a damped-driven stochastic Galerkin equation; a pointwise fractional-derivative inequality makes the dissipation rate of the energy coercive, producing stationary-measure bounds that are uniform in the damping and the Galerkin dimension. Third, an inviscid-infinite-dimensional-limit (IID-limit) procedure: stationary measures of the stochastic system converge, as the damping goes to zero, to invariant measures for the deterministic Galerkin flow; then a probabilistic representation principle plus Chebyshev-type tail estimates on the invariant measure selects a full-measure set of initial data whose Galerkin trajectories obey a uniform $(1+|t|)^\varepsilon$ growth bound. A globalization lemma converts those uniform bounds into global existence in $H^s$ for the limiting flow.

What would settle it

For a concrete admissible case, such as quintic NLS on the three-dimensional torus, compute (analytically or numerically) the limit measure $\mu$ obtained by the inviscid-infinite-dimensional procedure and check whether $\mu(\{u:\|u\|_{L^2}\le a\})\to 0$ as $a\to 0$; if this small-ball mass fails to vanish, or if an atom at any fixed $L^2$ value appears, then the full-measure ensemble of Theorem 1.1 cannot exist.

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Extended reading notes

Core claim

The central claim is that energy-supercritical NLS on compact manifolds is almost-surely globally well-posed for singular data. For every compact Riemannian manifold $(M^d,g)$ of dimension $d\ge 3$, every Sobolev order $s\in(s_{M^d},d/2]$, and every power nonlinearity $q\ge q_{M^d}$, the paper constructs a set $\Sigma=\Sigma_{q,s,\varepsilon}\subset H^s$ and a probability measure $\mu$ with the following properties: the solution map $\varphi^t$ is a global flow on $\Sigma$ with $\varphi^t\Sigma=\Sigma$; every trajectory satisfies $\|\varphi^t u_0\|_{H^{s-}}\le C(\|u_0\|_{H^s})(1+|t|)^\varepsilon$ for all $t$; $\mu(\Sigma)=1$; $\mu$ is invariant under $\varphi^t$; and the law of the functional $u\mapsto\|u\|_{L^2}$ under $\mu$ is absolutely continuous with respect to Lebesgue measure. On the torus and on Zoll manifolds the admissible regularity range extends to all $s>s_{q,d}=d/2-1/q$, the critical scaling exponent, while on general compact manifolds it is restricted to $s>s_{M^d}=d/2-1/(2q)$ by the available Strichartz estimates.

Load-bearing premise

The load-bearing premise is that the measure assigns negligible total weight to initial data whose overall size ($L^2$ norm) is small; the proof imports this small-ball negligibility from a cited local-time result, and without it the constructed set $\Sigma$ need not have full measure even if all trajectory bounds hold.

Editorial extensions

If this is right

For every compact Riemannian manifold of dimension $d\ge 3$ and every admissible singular regularity up to $d/2$, energy-supercritical NLS admits a global flow defined almost surely with respect to an invariant probability measure, so the supercritical obstruction is circumvented in a measure-theoretic sense.
The bound $\|\varphi^t u_0\|_{H^{s-}}\le C(1+|t|)^\varepsilon$ for every $\varepsilon>0$ gives quantitative long-time control: typical Sobolev norms grow at most like a very small power of time on compact manifolds, where scattering is absent.
The invariance of $\mu$ and the flow-invariance of the full-measure set $\Sigma$ imply recurrence-type behavior: typical trajectories return infinitely often to every set of positive measure, providing a statistical substitute for scattering on bounded domains.
On tori and Zoll manifolds the result covers all regularities above the critical exponent $s_{q,d}=d/2-1/q$, while on general compact manifolds it covers $s>s_{M^d}=s_{q,d}+1/(2q)$, widening the previously known range for singular data from balls to arbitrary compact geometries.
The absolute continuity of the $L^2$-norm distribution rules out atoms, in particular at $u=0$, so the invariant measure is not trapped on a single orbit or near the zero state.

Reading between the lines

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

The same ensemble construction could plausibly transfer to other energy-supercritical dispersive equations on compact domains (for instance nonlinear wave or Hartree equations) whenever a coercive dissipation operator and Strichartz estimates are available; the paper does not state this.
The dependence on small-ball negligibility of the $L^2$-norm distribution suggests a direct stress test: compute the stationary measures' small-ball mass for the cubic NLS on the 3-torus; failure of $\mu(\{u:\|u\|_{L^2}\le a\})\to0$ would break the full-measure property while leaving the trajectory bounds intact.
Because the growth bound holds for every $\varepsilon>0$, one might conjecture sharper sub-polynomial (e.g. logarithmic) growth for typical data; the present argument is only constructed to give the $\varepsilon$-polynomial bound.
The restriction $q\ge q_{M^d}$ stems from relying on linear Strichartz estimates; as the paper's remark indicates, multilinear refinement should lower the nonlinearity threshold, bringing more physical nonlinearities into the theorem's admissible range.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here is my read. The paper does something genuinely new: it pushes the IID-limit programme of Sy and Sy-Yu from tori and unit balls to general compact Riemannian manifolds, all dimensions, and singular Sobolev regularities s <= d/2 with all large nonlinearity powers. The local wellposedness argument based on Strichartz estimates is sensible, and the dissipation design using the Cordoba-Cordoba inequality is clever. If Theorem 1.1 is true, it is a significant advance in a genuinely hard supercritical regime. The soft spot the reader flagged is real and load-bearing. Lemma 4.9 tries to prove mu(Sigma_{s'})=1 from (4.16), which only controls mu_N(E_a^N \ Sigma^i_{N,s'}). To conclude mu_N(Sigma^i_{N,s'}) >= 1 - C i^{-2k} you need mu_N(E_a^N) close to 1, i.e. a small-ball estimate for mu_N(B_a). The paper has a large-mass tail (Prop 4.5) but no small-mass control, and the later non-atomicity of ||u||_{L2} under mu (Prop 4.13) is not uniform in N and comes after the fact. As written, the chain (4.26)-(4.29) is invalid. Since Sigma_{s'} is the set on which the globalization lemma applies, the almost-sure statement of Theorem 1.1 does not follow. This looks repairable, because the dissipative structure should give enough control near zero, but it has to be supplied. There is a second, smaller gap: Lemma 4.7 factorizes |u|^{2q}u - |v+z|^{2q}(v+z) using polynomials of degree 2q. That is fine for integer q, but the theorem allows real q, so for non-integer powers one needs a different estimate. This is likely fixable, but as written it is an unproven step. The text also has rough spots that make verification harder: typos, garbled norm definitions, and some skipped justifications. None of that changes the core, but it adds friction. Bottom line: the paper is worth serious refereeing and the program is credible. I would send it out with a request for major revision, focusing on the small-ball estimate and the power issue. As it stands, I would not cite the theorem without the repair.

Referee Report

3 major / 5 minor

Summary. The paper claims a probabilistic global well-posedness theorem (Theorem 1.1) for energy-supercritical nonlinear Schrödinger equations on compact Riemannian manifolds. The proof follows the IID-limit framework: a Galerkin system is stochastically damped and driven, stationary measures are constructed, the inviscid limit yields invariant measures μ_N for the Galerkin flow, and a statistical ensemble Σ is extracted on which uniform H^{s-} growth bounds permit a deterministic globalization argument. The main theorem asserts an invariant measure μ with μ(Σ)=1, flow invariance, almost-sure global existence for H^s data, polynomial-in-time growth, and absolute continuity of the L^2-mass distribution.

Significance. Should the proof be correct, this would be a substantial advance: it would extend almost-sure global well-posedness and invariant-measure constructions for energy-supercritical NLS from the torus and ball settings to general compact manifolds, cover all dimensions d≥3 and Sobolev orders up to H^{d/2}, and handle general nonlinearity powers. The explicit dissipation operator and the claimed simplification of the infinite-dimensional limit (Remark 1.3) are attractive features of the approach. However, the stated theorem is currently contingent on two load-bearing technical steps that are not justified as written: the full-measure property of the ensemble and the treatment of non-integer powers in the nonlinearity.

major comments (3)

[Lemma 4.9, Eq. (4.16)–(4.28)] The conclusion μ(Σ_{s'})=1 is not established. Proposition 4.8 provides only μ_N(E_a^N \setminus Σ_{i,N,s'}) ≤ C i^{-2k}, which controls failure outside B_a={‖u‖_{L^2}≤a}. In the chain (4.26)–(4.28), this is used as μ_N(Σ_{i,N,s'}) ≥ 1 - C i^{-2k}; that step requires μ_N(B_a) to be negligible uniformly in N. Proposition 4.5, Eq. (4.12), is a large-mass tail estimate and gives no control near the origin, while the later non-atomicity statement in Proposition 4.13(1) is not uniform in N, and no two-parameter limit (a→0, i→∞) is performed. Since Theorem 1.1(3) and the globalization Proposition 4.11 both depend on μ(Σ)=1, this gap is load-bearing.
[Lemma 4.7, Eq. (4.13)] The factorization |u|^{2q}u - |v+z|^{2q}(v+z) = w f_{2q}(u,v) - g_{2q}(v,z)z, with f_{2q} and g_{2q} described as polynomials of degree 2q, is an algebraic identity valid only when 2q is an integer. The theorem allows q≥q_{M^d} with non-integer values (e.g., on T^3, Corollary 2.4 gives q≥5/3), and Proposition 3.1 explicitly treats real q>1. This identity is used to prove the inviscid-limit convergence in Proposition 4.6, so the convergence (III) in that proof is not justified for non-integer q without a substitute estimate. The manuscript should either impose integrality of 2q (and of 3k̃, as needed for F∈C^∞ in Proposition 4.2) or prove the required Lipschitz/factorization bounds for real powers.
[Proposition 2.1 and Corollaries 2.3–2.5] The local theory for non-integer q needs clarification. Proposition 2.1 uses the estimate ‖|P_N u|^{2q}P_N u‖_{H^s} ≲ ‖P_N u‖_{L^∞}^{2q}‖P_N u‖_{H^s} for s up to d/2. For real, non-integer q, the map u↦|u|^{2q}u is not C^{⌈s⌉}-smooth at the origin when s>1, so the standard composition/product estimates used in the contraction argument require additional hypotheses or a separate fractional-calculus argument. Since the theorem claims all q≥q_{M^d}, the non-integer cases must be either excluded explicitly or handled by suitable fractional composition estimates; as written, the local well-posedness input is not fully justified in the stated generality.

minor comments (5)

[Section 4.4, Eq. (4.19)] The displayed inequality in (4.19) is tautological as written: the summand should involve μ_N(E_a^N ∩ φ_N^{-lT_0}(B_{i,j}^c)) (or an equivalent set), and the invariance/Chebyshev steps should be displayed explicitly.
[Lemma 4.9] The proof cites “the inequality (4.30) below” before Lemma 4.10 is stated; the order should be changed or the citation adjusted.
[Lemma 2.7] In the proof, several occurrences of W^{σ,q} should be W^{σ,p}; the Gronwall display and the surrounding estimates should be checked for consistency.
[References] Reference [58] is a duplicate of [57] with the same title, journal, volume, and pages; the bibliography should be cleaned up.
[Theorem 1.1] The notation s- is defined as s-ε in Section 1.7, but the theorem should state explicitly how ε in that notation is related to the growth exponent in (1.14).

Circularity Check

1 steps flagged · score 2.0 of 10

No circular derivation: the core estimates are self-contained, and the only same-author citation supports an ancillary measure property.

other [Section 4.6, Proposition 4.13(1)]
"The proof uses an argument of Shirikyan [49], and follows Theorem 9.1 in Sy [55]."

This is the single theorem-level property in Theorem 1.1 (item (5), absolute continuity of the L2-norm distribution) that is imported from the third author's prior work instead of being proved from the stochastic Galerkin system in this paper. It is not used in the main globalization argument: Lemma 4.9, Proposition 4.11, and the invariance proof in Proposition 4.13(2) do not rely on absolute continuity. Therefore this is a minor self-citation rather than a load-bearing circular reduction.

full rationale

The derivation chain for Theorem 1.1 is largely self-contained. The invariant measures mu_N are constructed from the damped-driven stochastic Galerkin system (4.1); the dissipation model in Section 3 is designed with explicit coercivity estimates, in particular (3.14); the uniform growth bounds in Proposition 4.8 and the Skorokhod/limsup argument in Lemma 4.9 do not fit any parameter to the target statement. The only same-author citation that carries a theorem property is Proposition 4.13(1), which defers the absolute-continuity statement to Theorem 9.1 in Sy [55]; that property is ancillary and is not needed for items (1)-(4) of Theorem 1.1. I therefore find no circular reduction. Separately, there is a non-circular correctness gap: inequality (4.16) only gives mu_N(Sigma^i_{N,s'}) >= mu_N(E_a^N) - C i^{-2k}, so the displayed chain (4.26)-(4.29) needs a uniform small-ball estimate mu_N(B_a) -> 0 that is not proved. Proposition 4.13(1) is proved later and only for the limit measure mu, so it does not repair the two-parameter limit needed in Lemma 4.9. This is a missing estimate, not an equivalence by construction, and is flagged as a correctness risk rather than as circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central result rests on the design of a strong polynomial damping, on an interpolation chain with parameters (C_{d,s}, ε, β), and on probabilistic machinery such as Skorokhod and Krylov-Bogoliubov arguments. The damping and noise coefficients are chosen to make the proof work, not fitted to data. The q-integer restriction is an unstated assumption. No new physical entities are posited apart from the artificial damping operator.

free parameters (5)

C_{d,s}
Coefficient of the polynomial damping term in L_s(u), Eq. (3.10); chosen large enough to absorb the high-order term in the energy dissipation rate (3.13)-(3.14).
ε (or k~=ε^{-1}) = k~ ≥ 2q/γ, ε small
Exponent in the damping ‖u‖^{3k~}_{H^{s-}}u; chosen small enough so that the interpolation and Young bounds in (3.14) work. It also controls the polynomial growth rate (1+|t|)^ε in Theorem 1.1.
β = β close to 0
Interpolation parameter in the s>2 case of the dissipation estimate; determines the required size of k~.
a (small-mass threshold) = arbitrary small
Defines E_a^N in Section 4.4; used to get a lower bound on L2 norm to control H^{s-} moments.
noise coefficients (a_n) = decay ensuring A0<∞ and A_{d/2-1/2}<∞
Define the covariance of the stochastic forcing in (4.1); the choice must satisfy two summability conditions.

assumptions (5)

domain assumption Strichartz estimates with loss on compact Riemannian manifolds, including (1.7) and refined torus/Zoll estimates (2.4)-(2.9).
Invoked in Proposition 2.1 and Corollaries 2.3-2.5; their validity is from [15,4,5,14] and is assumed as background.
standard math Cordoba-Cordoba inequality (3.3) for fractional Laplacian on the manifold.
Used in Proposition 3.1 to obtain lower bounds on the dissipation rate of energy; assumed as a known pointwise inequality.
standard math Sogge Lp eigenfunction estimates ‖e_n‖_{L∞} ≲ λ_n^{(d-1)/4}.
Used in Proposition 4.3 to control the second-order Itô correction for the energy; cited as [50].
ad hoc to paper The nonlinearity |u|^{2q}u is smooth enough for product and factorization estimates, i.e. 2q is an integer.
Lemma 4.7 writes the difference of nonlinearities as a polynomial identity of degree 2q; this is only valid for integer powers, but the paper only assumes q≥1.
standard math The data (u0,N) approximating u0 lie in EN and converge in Hs; Skorokhod representation and Ulam regularity hold.
Used in Sections 4.5-4.6 to transfer measures and prove invariance; standard probabilistic facts.

invented entities (1)

Dissipation operator L_s(u)=(-Δ)^{s-1}u + C_{d,s}‖u‖^{3k~}_{H^{s-}}u
purpose: Artificial damping in the stochastic Galerkin equation (4.1); designed so that the mass and energy dissipation rates are coercive uniformly in N and α.
This is a mathematical device, not a physical mechanism; no empirical prediction is attached to it.

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Pith. "Pith review of Probabilistic global-wellposedness for the energy-supercritical Schr\"odinger equations on compact manifolds." pith.science (2026). https://pith.science/paper/S6QKSZ53

@misc{pith2026250208812,
  author       = {Pith},
  title        = {Pith review of: Probabilistic global-wellposedness for the energy-supercritical Schr\"odinger equations on compact manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6QKSZ53}},
  note         = {Machine review of arXiv:2502.08812}
}

abstract

We consider the nonlinear Schr\"odinger equations with a general nonlinearity power in all dimensions. We construct invariant measures concentrated on Sobolev spaces $H^s$ of singular orders, $s\leq\frac{d}{2}$. We prove almost sure global wellposedness and bounds on the growth in time of the solutions via invariant measure arguments. Our setting includes a generic compact Riemannian manifold; we specify the cases of the torus and Zoll manifolds.

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Works this paper leans on

64 extracted references · 60 canonical work pages

[1]

R. Anton. Cubic nonlinear Schr¨ odinger equation on three dimensional balls with radial data. Comm. Partial Diﬀerential Equations, 33(10-12):1862–1889, 2008

work page 2008
[2]

R. Anton. Strichartz inequalities for lipschitz metric s on manifolds and nonlinear schr¨ odinger equation on domai ns. Bulletin de la Soci´ et´ e Math´ ematique de France, 136(1):27–65, 2008

work page 2008
[3]

Large global solutions for nonlinear Schr\"odinger equations III, energy-supercritical cases

M. Beceanu, Q. Deng, A. Soﬀer, and Y. W u. Large global solu tions for nonlinear Schr¨ odinger equations iii, energy- supercritical cases. arXiv preprint arXiv:1901.07709 , 2019

work page Pith review arXiv 1901
[4]

M. D. Blair, H. F. Smith, and C. D. Sogge. Strichartz estim ates and the nonlinear Schr¨ odinger equation on manifolds with boundary. Mathematische Annalen , 354(4):1397–1430, 2012

work page 2012
[5]

Bourgain

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

work page 1993
[6]

Bourgain

J. Bourgain. Periodic nonlinear Schr¨ odinger equation and invariant measures. Communications in Mathematical Physics, 166:1–26, 1994

work page 1994
[7]

Bourgain

J. Bourgain. Invariant measures for the 2D-defocusing n onlinear Schr¨ odinger equation.Comm. Math. Phys. , 176(2):421– 445, 1996

work page 1996
[8]

Bourgain

J. Bourgain. Reﬁnements of Strichartz’ inequality and a pplications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices , 1998

work page 1998

Show all 64 references

[9]

Bourgain

J. Bourgain. Global wellposedness of defocusing critic al nonlinear Schr¨ odinger equation in the radial case. Journal of the American Mathematical Society , 12(1):145–171, 1999

1999
[10]

Bourgain and A

J. Bourgain and A. Bulut. Invariant gibbs measure evolu tion for the radial nonlinear wave equation on the 3d ball. Journal of Functional Analysis , 266(4):2319–2340, 2014

2014
[11]

Bourgain and C

J. Bourgain and C. Demeter. The proof of the l2 decouplin g conjecture. Annals of mathematics , pages 351–389, 2015

2015
[12]

Brezis and T

H. Brezis and T. Gallouet. Nonlinear schr¨ odinger evol ution equations. Nonlinear Analysis , 4(4):677–681, 1980

1980
[13]

Bringmann, Y

B. Bringmann, Y. Deng, A. R. Nahmod, and H. Yue. Invarian t gibbs measures for the three dimensional cubic nonlinear wave equation. Inventiones mathematicae , pages 1–279, 2024

2024
[14]

N. Burq, P. G´ erard, and N. Tzvetkov. The Cauchy problem for the nonlinear Schr¨ odinger equation on a compact manifold. Journal of Nonlinear Mathematical Physics , 10(Suppl 1):12–27, 2003

2003
[15]

N. Burq, P. G´ erard, and N. Tzvetkov. Strichartz inequa lities and the nonlinear Schr¨ odinger equation on compact manifolds. American Journal of Mathematics , 126(3):569–605, 2004

2004
[16]

N. Burq, P. G´ erard, and N. Tzvetkov. Bilinear eigenfun ction estimates and the nonlinear Schr¨ odinger equation on surfaces. Invent. Math. , 159(1):187–223, 2005

2005
[17]

N. Burq, L. Thomann, and N. Tzvetkov. Remarks on the Gibb s measures for nonlinear dispersive equations. Ann. Fac. Sci. Toulouse Math. (6) , 27(3):527–597, 2018

2018
[18]

Cazenave and F

T. Cazenave and F. B. W eissler. The Cauchy problem for th e critical nonlinear Schr¨ odinger equation in H s. Nonlinear Analysis: Theory, Methods & Applications , 14(10):807–836, 1990

1990
[19]

F. M. Christ and M. I. W einstein. Dispersion of small amp litude solutions of the generalized Korteweg-de Vries equa tion. Journal of functional analysis , 100(1):87–109, 1991

1991
[20]

Colliander, M

J. Colliander, M. Keel, G. Staﬃlani, H. Takaoka, and T. T ao. Almost conservation laws and global rough solutions to a nonlinear Schr¨ odinger equation.Math. Res. Lett. , 9(5-6):659–682, 2002

2002
[21]

Colliander, M

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

2008
[22]

Colliander, M

J. Colliander, M. Keel, G. Staﬃlani, H. Takaoka, and T. T ao. Transfer of energy to high frequencies in the cubic defocusing nonlinear schr¨ odinger equation. Inventiones mathematicae , 181(1):39–113, 2010

2010
[23]

C´ ordoba and D

A. C´ ordoba and D. C´ ordoba. A pointwise estimate for fr actionary derivatives with applications to partial diﬀere ntial equations. Proceedings of the National Academy of Sciences , 100(26):15316–15317, 2003

2003
[24]

Da Prato and A

G. Da Prato and A. Debussche. Two-dimensional navier–s tokes equations driven by a space–time white noise. Journal of Functional Analysis , 196(1):180–210, 2002

2002
[25]

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

2024
[26]

B. Dodson. Global well-posedness and scattering for no nlinear Schr¨ odinger equations with algebraic nonlinearity when d = 2,3 and u0 is radial. Cambridge journal of mathematics , 7, 2019. PROBABILISTIC GWP FOR ENERGY SUPERCRITICAL NLS ON COMPACT M ANIFOLDS 28

2019
[27]

F¨ oldes and M

J. F¨ oldes and M. Sy. Invariant measures and global well posedness for the sqg equation. Archive for Rational Mechanics and Analysis , 241(1):187–230, 2021

2021
[28]

Foldes and M

J. Foldes and M. Sy. Almost sure global well-posedness f or 3D Euler equation and other ﬂuid dynamics models. arXiv preprint arXiv:2401.00332, 2023

2023 arXiv
[29]

Ginibre and G

J. Ginibre and G. Velo. The global Cauchy problem for the non linear Schr¨ odinger equation revisited. In Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, volume 2, pages 309–327. Elsevier, 1985

1985
[30]

M. G. Grillakis. On nonlinear Schr¨ odinger equations. Comm. Partial Diﬀerential Equations , 25(9-10):1827–1844, 2000

2000
[31]

Z. Hani, B. Pausader, N. Tzvetkov, and N. Visciglia. Mod iﬁed scattering for the cubic schr¨ odinger equation on prod uct spaces and applications. In Forum of mathematics, Pi , volume 3, page e4. Cambridge University Press, 2015

2015
[32]

S. Herr. The quintic nonlinear schr¨ odinger equation o n three-dimensional zoll manifolds. American Journal of Mathe- matics, 135(5):1271–1290, 2013

2013
[33]

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

2011
[34]

Ionescu, B

A. Ionescu, B. Pausader, and G. Staﬃlani. On the global w ell-posedness of energy-critical schr¨ odinger equations in curved spaces. Analysis & PDE , 5(4):705–746, 2012

2012
[35]

A. D. Ionescu and B. Pausader. The energy-critical defo cusing NLS on T3. Duke Math. J. , 161(8):1581–1612, 2012

2012
[36]

Ivanovici

O. Ivanovici. On the Schr¨ odinger equation outside str ictly convex obstacles. Analysis & PDE , 3(3):261–293, 2010

2010
[37]

Keel and T

M. Keel and T. Tao. Endpoint Strichartz estimates. American Journal of Mathematics , 120(5):955–980, 1998

1998
[38]

C. E. Kenig and F. Merle. Nondispersive radial solution s to energy supercritical non-linear wave equations, with applications. American journal of mathematics , 133(4):1029–1065, 2011

2011
[39]

Killip and M

R. Killip and M. Visan. Energy-Supercritical NLS: Crit ical ˙H s− bounds imply scattering. Communications in Partial Diﬀerential Equations , 35(6):945–987, 2010

2010
[40]

Kuksin and A

S. Kuksin and A. Shirikyan. Randomly forced CGL equatio n: stationary measures and the inviscid limit. Journal of Physics A: Mathematical and General , 37(12):3805, 2004

2004
[41]

Kuksin and A

S. Kuksin and A. Shirikyan. Mathematics of two-dimensional turbulence , volume 194. Cambridge University Press, 2012

2012
[42]

S. B. Kuksin. The eulerian limit for 2d statistical hydr odynamics. Journal of statistical physics , 115:469–492, 2004

2004
[43]

M. Latocca. Construction of high regularity invariant measures for the 2d Euler equations and remarks on the growth of the solutions. Communications in Partial Diﬀerential Equations , 48(1):22–53, 2023

2023
[44]

J. L. Lebowitz, H. A. Rose, and E. R. Speer. Statistical m echanics of the nonlinear Schr¨ odinger equation. Journal of statistical physics , 50(3):657–687, 1988

1988
[45]

Merle, P

F. Merle, P. Rapha¨ el, I. Rodnianski, and J. Szeftel. On blow up for the energy super critical defocusing nonlinear Schr¨ odinger equations.Inventiones mathematicae , pages 1–167, 2022

2022
[46]

A. R. Nahmod, T. Oh, L. Rey-Bellet, and G. Staﬃlani. Inva riant weighted wiener measures and almost sure global well-posedness for the periodic derivative nls. Journal of the European Mathematical Society , 14(4):1275–1330, 2012

2012
[47]

Pausader, N

B. Pausader, N. Tzvetkov, and X. W ang. Global regularit y for the energy-critical NLS on S3. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 31(2):315–338, 2014

2014
[48]

Ryckman and M

E. Ryckman and M. Visan. Global well-posedness and scat tering for the defocusing energy-critical nonlinear Schr¨odinger equation in r1+4. American journal of mathematics , 129(1):1–60, 2007

2007
[49]

Shirikyan

A. Shirikyan. Local times for solutions of the complex G inzburg–Landau equation and the inviscid limit. Journal of mathematical analysis and applications , 384(1):130–137, 2011

2011
[50]

C. D. Sogge. Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds. Journal of functional analysis , 77(1):123–138, 1988

1988
[51]

Staﬃlani and D

G. Staﬃlani and D. Tataru. Strichartz estimates for a Sc hr¨ odinger operator with nonsmooth coeﬃcients. Communica- tions in Partial Diﬀerential Equations , 27(7-8):1337–1372, 2002

2002
[52]

R. S. Strichartz. Restrictions of Fourier transforms t o quadratic surfaces and decay of solutions of wave equation s. 1977

1977
[53]

M. Sy. Invariant measure and long time behavior of regul ar solutions of the Benjamin–Ono equation. Analysis & PDE , 11(8):1841–1879, 2018

2018
[54]

M. Sy. Invariant measure and large time dynamics of the c ubic Klein–Gordon equation in 3D. Stochastics and Partial Diﬀerential Equations: Analysis and Computations , 7(3):379–416, 2019

2019
[55]

M. Sy. Almost sure global well-posedness for the energy supercritical schr¨ odinger equations.Journal de Math´ ematiques Pures et Appliqu´ ees, 154:108–145, 2021

2021
[56]

Sy and X

M. Sy and X. Yu. Almost sure global well-posedness for th e energy supercritical nls on the unit ball of R3. arXiv preprint arXiv:2007.00766, 2020

2007 arXiv
[58]

Sy and X

M. Sy and X. Yu. Global well-posedness and long-time beh avior of the fractional nls. Stochastics and Partial Diﬀerential Equations: Analysis and Computations , 10(4):1261–1317, 2022

2022
[59]

Sy and X

M. Sy and X. Yu. Global well-posedness for the cubic frac tional nls on the unit disk. Nonlinearity, 35(4):2020, 2022

2020
[60]

T. Tao. Global well-posedness and scattering for the hi gher-dimensional energy-critical nonlinear schr¨ odinger equation for radial data. New York J. Math , 11:57–80, 2005

2005
[61]

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

2006
[62]

Tzvetkov

N. Tzvetkov. Invariant measures for the nonlinear Schr ¨ odinger equation on the disc.Dyn. Partial Diﬀer. Equ. , 3(2):111– 160, 2006

2006
[63]

Tzvetkov

N. Tzvetkov. Invariant measures for the defocusing non linear Schr¨ odinger equation. Ann. Inst. Fourier (Grenoble) , 58(7):2543–2604, 2008

2008
[64]

M. Visan. The defocusing energy-critical nonlinear Sc hr¨ odinger equation in higher dimensions. 2007

2007
[65]

W.-M. W ang. Energy supercritical nonlinear Schr¨ odin ger equations: quasiperiodic solutions. 2016. PROBABILISTIC GWP FOR ENERGY SUPERCRITICAL NLS ON COMPACT M ANIFOLDS 29 Seynabou Gueye AIMS-Senegal KM2 Route de Joal P.O Box,1418, Mbour-Thi `es, S ´en´egal Email address : gu...

2016

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

R. Anton. Cubic nonlinear Schr¨ odinger equation on three dimensional balls with radial data. Comm. Partial Diﬀerential Equations, 33(10-12):1862–1889, 2008

work page 2008

[2] [2]

R. Anton. Strichartz inequalities for lipschitz metric s on manifolds and nonlinear schr¨ odinger equation on domai ns. Bulletin de la Soci´ et´ e Math´ ematique de France, 136(1):27–65, 2008

work page 2008

[3] [3]

Large global solutions for nonlinear Schr\"odinger equations III, energy-supercritical cases

M. Beceanu, Q. Deng, A. Soﬀer, and Y. W u. Large global solu tions for nonlinear Schr¨ odinger equations iii, energy- supercritical cases. arXiv preprint arXiv:1901.07709 , 2019

work page Pith review arXiv 1901

[4] [4]

M. D. Blair, H. F. Smith, and C. D. Sogge. Strichartz estim ates and the nonlinear Schr¨ odinger equation on manifolds with boundary. Mathematische Annalen , 354(4):1397–1430, 2012

work page 2012

[5] [5]

Bourgain

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

work page 1993

[6] [6]

Bourgain

J. Bourgain. Periodic nonlinear Schr¨ odinger equation and invariant measures. Communications in Mathematical Physics, 166:1–26, 1994

work page 1994

[7] [7]

Bourgain

J. Bourgain. Invariant measures for the 2D-defocusing n onlinear Schr¨ odinger equation.Comm. Math. Phys. , 176(2):421– 445, 1996

work page 1996

[8] [8]

Bourgain

J. Bourgain. Reﬁnements of Strichartz’ inequality and a pplications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices , 1998

work page 1998