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New periodic orbits in the planar equal-mass three-body problem
Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data
1. | Université Côte d'Azur, LJAD, 06100, France |
2. | Department of Mathematics and Hubei Province Key Laboratory of Mathematical Physics, Central China Normal University, Wuhan 430079, China |
3. | Université Côte d'Azur, LJAD, 06100, France |
The aim of this paper is to adapt the strategy in [
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
A. V. Balakrishnan,
Fractional powers of closed operators and the semigroups generated by them, Pacific. J. Math., 10 (1960), 419-437.
doi: 10.2140/pjm.1960.10.419. |
[3] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blow up for fractional NLS, J. Functional Analysis, 271 (2016), 2569-2603.
doi: 10.1016/j.jfa.2016.08.011. |
[4] |
W. Chen, C. Miao and X. Yao,
Dispersive estimates with geometry of finite type, Communications in Partial Differential Equations, 37 (2012), 479-510.
doi: 10.1080/03605302.2011.641053. |
[5] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[6] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems, 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[7] |
V. D. Dinh, On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces, preprint, arXiv: 1701.00852. |
[8] |
B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS,
Proc. Amer. Math. Soc., 145 (2017), 4859-4867, arXiv: 1611.04195.
doi: 10.1090/proc/13678. |
[9] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacian in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[10] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[11] |
Z. Guo and Y. Wang,
Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, J. Anal. Math., 124 (2014), 1-38.
doi: 10.1007/s11854-014-0025-6. |
[12] |
Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrodinger equation in the radial case, preprint, arXiv: 1310.6816. |
[13] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Communications on Pure and Applied Analysis, 14 (2015), 2265-2282.
doi: 10.3934/cpaa.2015.14.2265. |
[14] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. App. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[15] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[16] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $ L^2$-critical half-wave equations, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[17] |
N. Laskin, Fractional Schrödinger equation Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[18] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $ H^1$ solution for the nonlinear Schrödinger equation, J. Differ. Eqns., 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[19] |
E. M. Stein,
Harmonic Analysis: Real-Variable Theory, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 2000. |
[20] |
T. Tao,
On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial. Differ.Equ., 1 (2004), 1-47.
doi: 10.4310/DPDE.2004.v1.n1.a1. |
show all references
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
A. V. Balakrishnan,
Fractional powers of closed operators and the semigroups generated by them, Pacific. J. Math., 10 (1960), 419-437.
doi: 10.2140/pjm.1960.10.419. |
[3] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blow up for fractional NLS, J. Functional Analysis, 271 (2016), 2569-2603.
doi: 10.1016/j.jfa.2016.08.011. |
[4] |
W. Chen, C. Miao and X. Yao,
Dispersive estimates with geometry of finite type, Communications in Partial Differential Equations, 37 (2012), 479-510.
doi: 10.1080/03605302.2011.641053. |
[5] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[6] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems, 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[7] |
V. D. Dinh, On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces, preprint, arXiv: 1701.00852. |
[8] |
B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS,
Proc. Amer. Math. Soc., 145 (2017), 4859-4867, arXiv: 1611.04195.
doi: 10.1090/proc/13678. |
[9] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacian in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[10] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[11] |
Z. Guo and Y. Wang,
Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, J. Anal. Math., 124 (2014), 1-38.
doi: 10.1007/s11854-014-0025-6. |
[12] |
Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrodinger equation in the radial case, preprint, arXiv: 1310.6816. |
[13] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Communications on Pure and Applied Analysis, 14 (2015), 2265-2282.
doi: 10.3934/cpaa.2015.14.2265. |
[14] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. App. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[15] |
C. E. Kenig and F. Merle,
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[16] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $ L^2$-critical half-wave equations, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[17] |
N. Laskin, Fractional Schrödinger equation Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[18] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $ H^1$ solution for the nonlinear Schrödinger equation, J. Differ. Eqns., 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[19] |
E. M. Stein,
Harmonic Analysis: Real-Variable Theory, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 2000. |
[20] |
T. Tao,
On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial. Differ.Equ., 1 (2004), 1-47.
doi: 10.4310/DPDE.2004.v1.n1.a1. |
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