-
Previous Article
Exponential decay of Lebesgue numbers
- DCDS Home
- This Issue
-
Next Article
On stacked central configurations of the planar coorbital satellites problem
Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations
1. | Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States |
References:
[1] |
D. Benney and A. Newell, Random wave closures, Stud. Appl. Math., 48 (1969), 29-53. |
[2] |
D. Benney and P. Saffman, Nonlinear interactions of random waves in a dispersive medium, Proc. Roy. Soc. A, 289 (1966), 301-320.
doi: 10.1098/rspa.1966.0013. |
[3] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. |
[4] |
J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Int. Math. Research Notices, 1996, 277-304. |
[5] |
J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Research Notices, 1998, 253-283. |
[6] |
J. Bourgain, "Nonlinear Schrödinger Equations," in "Hyperbolic Equations and Frequency Interactions'' (eds. L. Caffarelli and W. E), IAS/Park City Mathematics Series, 5, AMS, Providence, RI, (1999), 3-157. |
[7] |
J. Bourgain, Global solutions of nonlinear Schrödinger equations, AMS Colloquium Publications, 46, AMS, Providence, RI, 1999. |
[8] |
J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348.
doi: 10.1007/BF02791265. |
[9] |
N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Mathematical Research Letters, 9 (2002), 323-335. |
[10] |
N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., 159 (2005), 187-223.
doi: 10.1007/s00222-004-0388-x. |
[11] |
F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori, preprint, (2008), arXiv:0809.4633. |
[12] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, CIMS, New York, AMS, Providence, RI, 2003. |
[13] |
J. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. of the American Math. Soc., 353 (2001), 3307-3325.
doi: 10.1090/S0002-9947-01-02760-X. |
[14] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.
doi: 10.1137/S0036141001384387. |
[15] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54. |
[16] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749 (electronic).
doi: 10.1090/S0894-0347-03-00421-1. |
[17] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.
doi: 10.1016/S0022-1236(03)00218-0. |
[18] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[19] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbb{R}^2$, Disc. and Cont. Dynam. Sys., 21 (2008), 665-686.
doi: 10.3934/dcds.2008.21.665. |
[20] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math. (2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[21] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113.
doi: 10.1007/s00222-010-0242-2. |
[22] |
J.-M. Delort, Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds, preprint, International Mathematics Research Notices, 2010, 2305-2328. |
[23] |
B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d \geq 3$, preprint, (2009), arXiv:0912.2467. |
[24] |
B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=2$, preprint, (2010), arXiv:1006.1365. |
[25] |
B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=1$, preprint, (2009), arXiv:1010.0040. |
[26] |
J. Duoandikoetxea, "Fourier Analysis," Graduate Studies in Mathematics, 29, AMS, Providence, RI, 2001. |
[27] |
J. Fröhlich and E. Lenzmann, "Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation," Séminaire: É.D.P. 2003-2004, Exposé No. XIX, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004), 26 pp. |
[28] |
J. Ginibre and T. Ozawa, Long-range scattering for non-linear Schrödinger and Hartree equations in space dimension $n\geq 2$, Comm. Math. Phys., 151 (1993), 619-645.
doi: 10.1007/BF02097031. |
[29] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[30] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.
doi: 10.1007/BF01214768. |
[31] |
J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, Rev. Math. Phys., 12 (2000), 361-429.
doi: 10.1142/S0129055X00000137. |
[32] |
J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. II, Ann. H. P., 1 (2000), 753-800. |
[33] |
J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions, J. Diff. Eq., 175 (2001), 415-501.
doi: 10.1006/jdeq.2000.3969. |
[34] |
A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations, preprint, 2006. |
[35] | |
[36] |
N. Hayashi, P. Naumkin and T. Ozawa, "Scattering Theory for the Hartree Equation," Hokkaido University Preprints, Series 358, Nov., 1996. |
[37] |
C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. |
[38] |
C. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation, Transactions of the AMS, 348 (1996), 3323-3353.
doi: 10.1090/S0002-9947-96-01645-5. |
[39] |
C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, preprint, (2011), arXiv:1104.1229. |
[40] |
C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627.
doi: 10.1016/j.jfa.2007.09.008. |
[41] |
C. Miao, G. Xu and L. Zhao, The Cauchy problem for the Hartree equation, J. PDEs, 21 (2008), 22-24. |
[42] |
C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering, and blow-up for the energy critical, focusing Hartree equation in the radial case, Coll. Math., 114 (2009), 213-236.
doi: 10.4064/cm114-2-5. |
[43] |
C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$ -subcritical Hartree equation on $\mathbbmathbb{R}^{d}$, Ann. I. H. Poincaré, NA, 26 (2009), 1831-1852. |
[44] |
C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbbR^{1+n}$, Comm. PDEs, 36 (2011), 729-776.
doi: 10.1080/03605302.2010.531073. |
[45] |
C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure. Appl. Math., 25 (1972), 1-31.
doi: 10.1002/cpa.3160250103. |
[46] |
B. Schlein, "Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics," Lecture Notes, Clay Summer School on Evolution Equations, Zurich, (2008), arXiv:0807.4307. |
[47] |
C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. Jour., 53 (1986), 43-65.
doi: 10.1215/S0012-7094-86-05303-2. |
[48] |
C. Sogge, Concerning the $\ell^p$ norm of spectral clusters for second order elliptic operators on compact manifolds, Jour. of Funct. Anal., 77 (1988), 123-138.
doi: 10.1016/0022-1236(88)90081-X. |
[49] |
V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrödinger Equations on $S^1$, to appear in Diff. and Int. Eqs., (2010), arXiv:1003.5705. |
[50] |
V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb{R}$, to appear in Indiana Univ. Math. J., (2010), arXiv:1003.5707. |
[51] |
G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142.
doi: 10.1215/S0012-7094-97-08604-X. |
[52] |
G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Math. J., 86 (1997), 79-107.
doi: 10.1215/S0012-7094-97-08603-8. |
[53] |
T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS Reg. Conf. Series in Math., 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, AMS, Providence, RI, 2006. |
[54] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on a surface of deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194. |
[55] |
S.-J. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds, J. Differential Equations, 245 (2008), 359-376.
doi: 10.1016/j.jde.2008.03.008. |
show all references
References:
[1] |
D. Benney and A. Newell, Random wave closures, Stud. Appl. Math., 48 (1969), 29-53. |
[2] |
D. Benney and P. Saffman, Nonlinear interactions of random waves in a dispersive medium, Proc. Roy. Soc. A, 289 (1966), 301-320.
doi: 10.1098/rspa.1966.0013. |
[3] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. |
[4] |
J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Int. Math. Research Notices, 1996, 277-304. |
[5] |
J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Research Notices, 1998, 253-283. |
[6] |
J. Bourgain, "Nonlinear Schrödinger Equations," in "Hyperbolic Equations and Frequency Interactions'' (eds. L. Caffarelli and W. E), IAS/Park City Mathematics Series, 5, AMS, Providence, RI, (1999), 3-157. |
[7] |
J. Bourgain, Global solutions of nonlinear Schrödinger equations, AMS Colloquium Publications, 46, AMS, Providence, RI, 1999. |
[8] |
J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348.
doi: 10.1007/BF02791265. |
[9] |
N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Mathematical Research Letters, 9 (2002), 323-335. |
[10] |
N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., 159 (2005), 187-223.
doi: 10.1007/s00222-004-0388-x. |
[11] |
F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori, preprint, (2008), arXiv:0809.4633. |
[12] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, CIMS, New York, AMS, Providence, RI, 2003. |
[13] |
J. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. of the American Math. Soc., 353 (2001), 3307-3325.
doi: 10.1090/S0002-9947-01-02760-X. |
[14] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.
doi: 10.1137/S0036141001384387. |
[15] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54. |
[16] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749 (electronic).
doi: 10.1090/S0894-0347-03-00421-1. |
[17] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.
doi: 10.1016/S0022-1236(03)00218-0. |
[18] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[19] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbb{R}^2$, Disc. and Cont. Dynam. Sys., 21 (2008), 665-686.
doi: 10.3934/dcds.2008.21.665. |
[20] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math. (2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[21] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113.
doi: 10.1007/s00222-010-0242-2. |
[22] |
J.-M. Delort, Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds, preprint, International Mathematics Research Notices, 2010, 2305-2328. |
[23] |
B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d \geq 3$, preprint, (2009), arXiv:0912.2467. |
[24] |
B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=2$, preprint, (2010), arXiv:1006.1365. |
[25] |
B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=1$, preprint, (2009), arXiv:1010.0040. |
[26] |
J. Duoandikoetxea, "Fourier Analysis," Graduate Studies in Mathematics, 29, AMS, Providence, RI, 2001. |
[27] |
J. Fröhlich and E. Lenzmann, "Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation," Séminaire: É.D.P. 2003-2004, Exposé No. XIX, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004), 26 pp. |
[28] |
J. Ginibre and T. Ozawa, Long-range scattering for non-linear Schrödinger and Hartree equations in space dimension $n\geq 2$, Comm. Math. Phys., 151 (1993), 619-645.
doi: 10.1007/BF02097031. |
[29] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[30] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.
doi: 10.1007/BF01214768. |
[31] |
J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, Rev. Math. Phys., 12 (2000), 361-429.
doi: 10.1142/S0129055X00000137. |
[32] |
J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. II, Ann. H. P., 1 (2000), 753-800. |
[33] |
J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions, J. Diff. Eq., 175 (2001), 415-501.
doi: 10.1006/jdeq.2000.3969. |
[34] |
A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations, preprint, 2006. |
[35] | |
[36] |
N. Hayashi, P. Naumkin and T. Ozawa, "Scattering Theory for the Hartree Equation," Hokkaido University Preprints, Series 358, Nov., 1996. |
[37] |
C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. |
[38] |
C. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation, Transactions of the AMS, 348 (1996), 3323-3353.
doi: 10.1090/S0002-9947-96-01645-5. |
[39] |
C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, preprint, (2011), arXiv:1104.1229. |
[40] |
C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627.
doi: 10.1016/j.jfa.2007.09.008. |
[41] |
C. Miao, G. Xu and L. Zhao, The Cauchy problem for the Hartree equation, J. PDEs, 21 (2008), 22-24. |
[42] |
C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering, and blow-up for the energy critical, focusing Hartree equation in the radial case, Coll. Math., 114 (2009), 213-236.
doi: 10.4064/cm114-2-5. |
[43] |
C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$ -subcritical Hartree equation on $\mathbbmathbb{R}^{d}$, Ann. I. H. Poincaré, NA, 26 (2009), 1831-1852. |
[44] |
C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbbR^{1+n}$, Comm. PDEs, 36 (2011), 729-776.
doi: 10.1080/03605302.2010.531073. |
[45] |
C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure. Appl. Math., 25 (1972), 1-31.
doi: 10.1002/cpa.3160250103. |
[46] |
B. Schlein, "Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics," Lecture Notes, Clay Summer School on Evolution Equations, Zurich, (2008), arXiv:0807.4307. |
[47] |
C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. Jour., 53 (1986), 43-65.
doi: 10.1215/S0012-7094-86-05303-2. |
[48] |
C. Sogge, Concerning the $\ell^p$ norm of spectral clusters for second order elliptic operators on compact manifolds, Jour. of Funct. Anal., 77 (1988), 123-138.
doi: 10.1016/0022-1236(88)90081-X. |
[49] |
V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrödinger Equations on $S^1$, to appear in Diff. and Int. Eqs., (2010), arXiv:1003.5705. |
[50] |
V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb{R}$, to appear in Indiana Univ. Math. J., (2010), arXiv:1003.5707. |
[51] |
G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142.
doi: 10.1215/S0012-7094-97-08604-X. |
[52] |
G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Math. J., 86 (1997), 79-107.
doi: 10.1215/S0012-7094-97-08603-8. |
[53] |
T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS Reg. Conf. Series in Math., 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, AMS, Providence, RI, 2006. |
[54] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on a surface of deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194. |
[55] |
S.-J. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds, J. Differential Equations, 245 (2008), 359-376.
doi: 10.1016/j.jde.2008.03.008. |
[1] |
Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure and Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341 |
[2] |
Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005 |
[3] |
Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $ h\mathbb{Z} $. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131 |
[4] |
F. Catoire, W. M. Wang. Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Communications on Pure and Applied Analysis, 2010, 9 (2) : 483-491. doi: 10.3934/cpaa.2010.9.483 |
[5] |
Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 |
[6] |
Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637 |
[7] |
Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072 |
[8] |
Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431 |
[9] |
François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229 |
[10] |
J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka, T. Tao. Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 665-686. doi: 10.3934/dcds.2008.21.665 |
[11] |
Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066 |
[12] |
Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037 |
[13] |
Yingying Xie, Jian Su, Liquan Mei. Blowup results and concentration in focusing Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5001-5017. doi: 10.3934/dcds.2020209 |
[14] |
D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 |
[15] |
Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure and Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 |
[16] |
Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905 |
[17] |
Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235 |
[18] |
Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 |
[19] |
Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 |
[20] |
Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]