---
_id: '8497'
abstract:
- lang: eng
  text: "We study the dynamics of the restricted planar three-body problem near mean
    motion resonances, i.e. a resonance involving the Keplerian periods of the two
    lighter bodies revolving around the most massive one. This problem is often used
    to model Sun–Jupiter–asteroid systems. For the primaries (Sun and Jupiter), we
    pick a realistic mass ratio μ=10−3 and a small eccentricity e0>0. The main result
    is a construction of a variety of non local diffusing orbits which show a drastic
    change of the osculating (instant) eccentricity of the asteroid, while the osculating
    semi major axis is kept almost constant. The proof relies on the careful analysis
    of the circular problem, which has a hyperbolic structure, but for which diffusion
    is prevented by KAM tori. In the proof we verify certain non-degeneracy conditions
    numerically.\r\n\r\nBased on the work of Treschev, it is natural to conjecture
    that the time of diffusion for this problem is ∼−ln(μe0)μ3/2e0. We expect our
    instability mechanism to apply to realistic values of e0 and we give heuristic
    arguments in its favor. If so, the applicability of Nekhoroshev theory to the
    three-body problem as well as the long time stability become questionable.\r\n\r\nIt
    is well known that, in the Asteroid Belt, located between the orbits of Mars and
    Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly
    at mean motion resonances of low order. Our mechanism gives a possible explanation
    of their existence. To relate the existence of Kirkwood gaps with Arnol'd diffusion,
    we also state a conjecture on its existence for a typical ϵ-perturbation of the
    product of the pendulum and the rotator. Namely, we predict that a positive conditional
    measure of initial conditions concentrated in the main resonance exhibits Arnol’d
    diffusion on time scales −lnϵϵ2."
article_processing_charge: No
article_type: original
author:
- first_name: Jacques
  full_name: Féjoz, Jacques
  last_name: Féjoz
- first_name: Marcel
  full_name: Guàrdia, Marcel
  last_name: Guàrdia
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Pablo
  full_name: Roldán, Pablo
  last_name: Roldán
citation:
  ama: Féjoz J, Guàrdia M, Kaloshin V, Roldán P. Kirkwood gaps and diffusion along
    mean motion resonances in the restricted planar three-body problem. <i>Journal
    of the European Mathematical Society</i>. 2016;18(10):2315-2403. doi:<a href="https://doi.org/10.4171/jems/642">10.4171/jems/642</a>
  apa: Féjoz, J., Guàrdia, M., Kaloshin, V., &#38; Roldán, P. (2016). Kirkwood gaps
    and diffusion along mean motion resonances in the restricted planar three-body
    problem. <i>Journal of the European Mathematical Society</i>. European Mathematical
    Society Publishing House. <a href="https://doi.org/10.4171/jems/642">https://doi.org/10.4171/jems/642</a>
  chicago: Féjoz, Jacques, Marcel Guàrdia, Vadim Kaloshin, and Pablo Roldán. “Kirkwood
    Gaps and Diffusion along Mean Motion Resonances in the Restricted Planar Three-Body
    Problem.” <i>Journal of the European Mathematical Society</i>. European Mathematical
    Society Publishing House, 2016. <a href="https://doi.org/10.4171/jems/642">https://doi.org/10.4171/jems/642</a>.
  ieee: J. Féjoz, M. Guàrdia, V. Kaloshin, and P. Roldán, “Kirkwood gaps and diffusion
    along mean motion resonances in the restricted planar three-body problem,” <i>Journal
    of the European Mathematical Society</i>, vol. 18, no. 10. European Mathematical
    Society Publishing House, pp. 2315–2403, 2016.
  ista: Féjoz J, Guàrdia M, Kaloshin V, Roldán P. 2016. Kirkwood gaps and diffusion
    along mean motion resonances in the restricted planar three-body problem. Journal
    of the European Mathematical Society. 18(10), 2315–2403.
  mla: Féjoz, Jacques, et al. “Kirkwood Gaps and Diffusion along Mean Motion Resonances
    in the Restricted Planar Three-Body Problem.” <i>Journal of the European Mathematical
    Society</i>, vol. 18, no. 10, European Mathematical Society Publishing House,
    2016, pp. 2315–403, doi:<a href="https://doi.org/10.4171/jems/642">10.4171/jems/642</a>.
  short: J. Féjoz, M. Guàrdia, V. Kaloshin, P. Roldán, Journal of the European Mathematical
    Society 18 (2016) 2315–2403.
date_created: 2020-09-18T10:46:31Z
date_published: 2016-09-19T00:00:00Z
date_updated: 2021-01-12T08:19:41Z
day: '19'
doi: 10.4171/jems/642
extern: '1'
intvolume: '        18'
issue: '10'
language:
- iso: eng
month: '09'
oa_version: None
page: 2315-2403
publication: Journal of the European Mathematical Society
publication_identifier:
  issn:
  - 1435-9855
publication_status: published
publisher: European Mathematical Society Publishing House
quality_controlled: '1'
status: public
title: Kirkwood gaps and diffusion along mean motion resonances in the restricted
  planar three-body problem
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 18
year: '2016'
...
---
_id: '8499'
abstract:
- lang: eng
  text: "We consider the cubic defocusing nonlinear Schrödinger equation in the two
    dimensional torus. Fix s>1. Recently Colliander, Keel, Staffilani, Tao and Takaoka
    proved the existence of solutions with s-Sobolev norm growing in time.\r\n\r\nWe
    establish the existence of solutions with polynomial time estimates. More exactly,
    there is c>0 such that for any K≫1 we find a solution u and a time T such that
    ∥u(T)∥Hs≥K∥u(0)∥Hs. Moreover, the time T satisfies the polynomial bound 0<T<Kc."
article_processing_charge: No
article_type: original
author:
- first_name: Marcel
  full_name: Guardia, Marcel
  last_name: Guardia
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
citation:
  ama: Guardia M, Kaloshin V. Growth of Sobolev norms in the cubic defocusing nonlinear
    Schrödinger equation. <i>Journal of the European Mathematical Society</i>. 2015;17(1):71-149.
    doi:<a href="https://doi.org/10.4171/jems/499">10.4171/jems/499</a>
  apa: Guardia, M., &#38; Kaloshin, V. (2015). Growth of Sobolev norms in the cubic
    defocusing nonlinear Schrödinger equation. <i>Journal of the European Mathematical
    Society</i>. European Mathematical Society Publishing House. <a href="https://doi.org/10.4171/jems/499">https://doi.org/10.4171/jems/499</a>
  chicago: Guardia, Marcel, and Vadim Kaloshin. “Growth of Sobolev Norms in the Cubic
    Defocusing Nonlinear Schrödinger Equation.” <i>Journal of the European Mathematical
    Society</i>. European Mathematical Society Publishing House, 2015. <a href="https://doi.org/10.4171/jems/499">https://doi.org/10.4171/jems/499</a>.
  ieee: M. Guardia and V. Kaloshin, “Growth of Sobolev norms in the cubic defocusing
    nonlinear Schrödinger equation,” <i>Journal of the European Mathematical Society</i>,
    vol. 17, no. 1. European Mathematical Society Publishing House, pp. 71–149, 2015.
  ista: Guardia M, Kaloshin V. 2015. Growth of Sobolev norms in the cubic defocusing
    nonlinear Schrödinger equation. Journal of the European Mathematical Society.
    17(1), 71–149.
  mla: Guardia, Marcel, and Vadim Kaloshin. “Growth of Sobolev Norms in the Cubic
    Defocusing Nonlinear Schrödinger Equation.” <i>Journal of the European Mathematical
    Society</i>, vol. 17, no. 1, European Mathematical Society Publishing House, 2015,
    pp. 71–149, doi:<a href="https://doi.org/10.4171/jems/499">10.4171/jems/499</a>.
  short: M. Guardia, V. Kaloshin, Journal of the European Mathematical Society 17
    (2015) 71–149.
date_created: 2020-09-18T10:46:50Z
date_published: 2015-02-05T00:00:00Z
date_updated: 2021-01-12T08:19:41Z
day: '05'
doi: 10.4171/jems/499
extern: '1'
intvolume: '        17'
issue: '1'
language:
- iso: eng
month: '02'
oa_version: None
page: 71-149
publication: Journal of the European Mathematical Society
publication_identifier:
  issn:
  - 1435-9855
publication_status: published
publisher: European Mathematical Society Publishing House
quality_controlled: '1'
status: public
title: Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 17
year: '2015'
...