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August  2021, 41(8): 3973-3984. doi: 10.3934/dcds.2021024

Decay estimates for nonlinear Schrödinger equations

Chenjie Fan 1, and  Zehua Zhao 2,

1. 

Academy of Mathematics and Systems Science, CAS, China
2. 

University of Maryland, USA

Received  August 2020 Revised  December 2020 Published  August 2021 Early access  January 2021

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).

Keywords: Nonlinear Schrödinger equation, decay estimate.
Mathematics Subject Classification: 35Q55, 35R01, 37K06, 37L50.
Citation: Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024
References:
[1]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10, American Mathematical Soc., 2003. doi: 10.1090/cln/010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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B. Dodson, Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space, arXiv preprint, arXiv: 2004.09618. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, vol. 106, American Mathematical Soc., 2006. doi: 10.1090/cbms/106.  Google Scholar

[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. Google Scholar

show all references

References:
[1]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10, American Mathematical Soc., 2003. doi: 10.1090/cln/010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, vol. 106, American Mathematical Soc., 2006. doi: 10.1090/cbms/106.  Google Scholar

[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. Google Scholar

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