Decay estimates for nonlinear Schrödinger equations

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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 and 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.
[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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