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Decay estimates for nonlinear Schrödinger equations
1. | Academy of Mathematics and Systems Science, CAS, China |
2. | University of Maryland, USA |
In this short note, we present some decay estimates for nonlinear solutions of 3d quintic, 3d cubic NLS, and 2d quintic NLS (nonlinear Schrödinger equations).
References:
[1] |
T. Cazenave, Semilinear Schrödinger Equations, vol. 10, American Mathematical Soc., 2003.
doi: 10.1090/cln/010. |
[2] |
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., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global existence and scattering for rough solutions of a nonlinear schrödinger equation on $\mathbb{R}^{3}$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
doi: 10.1002/cpa.20029. |
[4] |
B. Dodson, Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space, arXiv preprint, arXiv: 2004.09618. |
[5] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN, 2009 (2009), 414-432.
doi: 10.1093/imrn/rnn135. |
[6] |
M. Grillakis and M. Machedon,
Pair excitations and the mean field approximation of interacting bosons, I, Comm. Math. Phys., 324 (2013), 601-636.
doi: 10.1007/s00220-013-1818-7. |
[7] |
N. Hayashi and M. Tsutsumi,
$L^\infty(\mathbf{R}^n)$-decay of classical solutions for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 309-327.
doi: 10.1017/S0308210500019235. |
[8] |
J.-L. Journé, A. Soffer and C. D. Sogge,
Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.
doi: 10.1002/cpa.3160440504. |
[9] |
C. Kenig and F. Merle,
Scattering for $\dot{H}^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Transactions of the American Mathematical Society, 362 (2010), 1937-1962.
doi: 10.1090/S0002-9947-09-04722-9. |
[10] |
S. Klainerman,
Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332.
doi: 10.1002/cpa.3160380305. |
[11] |
S. Klainerman and G. Ponce,
Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math., 36 (1983), 133-141.
doi: 10.1002/cpa.3160360106. |
[12] |
J.-E. Lin and W. A. Strauss,
Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Functional Analysis, 30 (1978), 245-263.
doi: 10.1016/0022-1236(78)90073-3. |
[13] |
F. Planchon and L. Vega,
Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér., 42 (2009), 261-290.
doi: 10.24033/asens.2096. |
[14] |
J. Shatah,
Global existence of small solutions to nonlinear evolution equations, J. Differential Equations, 46 (1982), 409-425.
doi: 10.1016/0022-0396(82)90102-4. |
[15] |
T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, vol. 106, American Mathematical Soc., 2006.
doi: 10.1090/cbms/106. |
[16] |
X. Yu, Global well-posedness and scattering for the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in $\mathbb{R}^{2}$, arXiv preprint, arXiv: 1805.03230. |
show all references
References:
[1] |
T. Cazenave, Semilinear Schrödinger Equations, vol. 10, American Mathematical Soc., 2003.
doi: 10.1090/cln/010. |
[2] |
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., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global existence and scattering for rough solutions of a nonlinear schrödinger equation on $\mathbb{R}^{3}$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
doi: 10.1002/cpa.20029. |
[4] |
B. Dodson, Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space, arXiv preprint, arXiv: 2004.09618. |
[5] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN, 2009 (2009), 414-432.
doi: 10.1093/imrn/rnn135. |
[6] |
M. Grillakis and M. Machedon,
Pair excitations and the mean field approximation of interacting bosons, I, Comm. Math. Phys., 324 (2013), 601-636.
doi: 10.1007/s00220-013-1818-7. |
[7] |
N. Hayashi and M. Tsutsumi,
$L^\infty(\mathbf{R}^n)$-decay of classical solutions for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 309-327.
doi: 10.1017/S0308210500019235. |
[8] |
J.-L. Journé, A. Soffer and C. D. Sogge,
Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.
doi: 10.1002/cpa.3160440504. |
[9] |
C. Kenig and F. Merle,
Scattering for $\dot{H}^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Transactions of the American Mathematical Society, 362 (2010), 1937-1962.
doi: 10.1090/S0002-9947-09-04722-9. |
[10] |
S. Klainerman,
Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332.
doi: 10.1002/cpa.3160380305. |
[11] |
S. Klainerman and G. Ponce,
Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math., 36 (1983), 133-141.
doi: 10.1002/cpa.3160360106. |
[12] |
J.-E. Lin and W. A. Strauss,
Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Functional Analysis, 30 (1978), 245-263.
doi: 10.1016/0022-1236(78)90073-3. |
[13] |
F. Planchon and L. Vega,
Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér., 42 (2009), 261-290.
doi: 10.24033/asens.2096. |
[14] |
J. Shatah,
Global existence of small solutions to nonlinear evolution equations, J. Differential Equations, 46 (1982), 409-425.
doi: 10.1016/0022-0396(82)90102-4. |
[15] |
T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, vol. 106, American Mathematical Soc., 2006.
doi: 10.1090/cbms/106. |
[16] |
X. Yu, Global well-posedness and scattering for the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in $\mathbb{R}^{2}$, arXiv preprint, arXiv: 1805.03230. |
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