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May 2019, 39(5): 2961-2976. doi: 10.3934/dcds.2019123

Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ

Abdelwahab Bensouilah and  Sahbi Keraani

Laboratoire Paul Painlevé (U.M.R. CNRS 8524), U.F.R. de Mathématiques, Université Lille 1, 59655 Villeneuve d'Ascq Cedex, France

Received  October 2018 Published  January 2019

Fund Project: This research was supported by Labex CEMPI (ANR-11-LABX-0007-01)

The paper reconsiders the issue of the regularity of the Duhamel part of the solution to the $ L^2 $-critical high-order NLS already studied by the authors in [4]. This model includes the mass critical 4D fourth order NLS. The improvement is due to the use of a more sophisticated space involving a nonlinear term and taking profit of a suitable trilinear Strichartz estimate.

Keywords: High-order NLS, smoothing effect, trilinear Strichartz estimates.
Mathematics Subject Classification: 35Q55, 35B40, 35B05.
Citation: Abdelwahab Bensouilah, Sahbi Keraani. Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123
References:
[1]

H. Bahouri, J-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

P. Bégout and A. Vargas, Mass concentration phenomena for the L2-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X.

[3]

M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Math. Acad. Sci., Paris, 330 (2000), 87–92. doi: 10.1016/S0764-4442(00)00120-8.

[4]

A. Bensouilah and S. Keraani, Smoothing property for the mass critical high-order Schrödinger equation Ⅰ, preprint, April 2018, submitted.
[5]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246. doi: 10.24033/asens.1404.

[6]

J. Bourgain, Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not., (1998), 253-283. doi: 10.1155/S1073792898000191.

[7]

M. Chae, S. Hong and S. Lee, Mass concentration for the L2-critical nonlinear Schrödinger equations of higher orders, Discrete Contin. Dyn. Syst., 29 (2011), 909-928. doi: 10.3934/dcds.2011.29.909.

[8]

P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, Journal of the American Mathematical Society, 1 (1988), 413-439. doi: 10.1090/S0894-0347-1988-0928265-0.

[9]

G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241.

[10]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.

[11]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys D., 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.

[12]

S. Keraani and A. Vargas, A smoothing property for the $L^2$-critical NLS equations and an application to blowup theory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 745-762. doi: 10.1016/j.anihpc.2008.03.001.

[13]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2009.

[14]

T. Oh and N. Tzvetkov, Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Related Fields, 169 (2017), 1121-1168. doi: 10.1007/s00440-016-0748-7.

[15]

____, On the transport of Gaussian measures under the flow of Hamiltonian PDEs, Sémin. Equ. Dériv. Partielles. 2015-2016, Exp. No. 6, 9 pp.
[16]

T. Oh, N. Tzvetkov and Y. Wang, Solving the 4NLS with white noise initial data, preprint.
[17]

B. Pausader, Global well-posedness for energy criticalfourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.

[18]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.

[19]

B. Pausader and S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyp. Differ. Equ., 7 (2010), 651-705. doi: 10.1142/S0219891610002256.

[20]

P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J., 55 (1987), 699-715. doi: 10.1215/S0012-7094-87-05535-9.

[21]

L. Vega, Schrödinger equation: Pointwise convergence to the initial data, Proc. Am. Math. Soc., 102 (1988), 874-878. doi: 10.2307/2047326.

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, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

P. Bégout and A. Vargas, Mass concentration phenomena for the L2-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X.

[3]

M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Math. Acad. Sci., Paris, 330 (2000), 87–92. doi: 10.1016/S0764-4442(00)00120-8.

[4]

A. Bensouilah and S. Keraani, Smoothing property for the mass critical high-order Schrödinger equation Ⅰ, preprint, April 2018, submitted.
[5]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246. doi: 10.24033/asens.1404.

[6]

J. Bourgain, Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not., (1998), 253-283. doi: 10.1155/S1073792898000191.

[7]

M. Chae, S. Hong and S. Lee, Mass concentration for the L2-critical nonlinear Schrödinger equations of higher orders, Discrete Contin. Dyn. Syst., 29 (2011), 909-928. doi: 10.3934/dcds.2011.29.909.

[8]

P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, Journal of the American Mathematical Society, 1 (1988), 413-439. doi: 10.1090/S0894-0347-1988-0928265-0.

[9]

G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241.

[10]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.

[11]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys D., 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.

[12]

S. Keraani and A. Vargas, A smoothing property for the $L^2$-critical NLS equations and an application to blowup theory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 745-762. doi: 10.1016/j.anihpc.2008.03.001.

[13]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2009.

[14]

T. Oh and N. Tzvetkov, Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Related Fields, 169 (2017), 1121-1168. doi: 10.1007/s00440-016-0748-7.

[15]

____, On the transport of Gaussian measures under the flow of Hamiltonian PDEs, Sémin. Equ. Dériv. Partielles. 2015-2016, Exp. No. 6, 9 pp.
[16]

T. Oh, N. Tzvetkov and Y. Wang, Solving the 4NLS with white noise initial data, preprint.
[17]

B. Pausader, Global well-posedness for energy criticalfourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.

[18]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.

[19]

B. Pausader and S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyp. Differ. Equ., 7 (2010), 651-705. doi: 10.1142/S0219891610002256.

[20]

P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J., 55 (1987), 699-715. doi: 10.1215/S0012-7094-87-05535-9.

[21]

L. Vega, Schrödinger equation: Pointwise convergence to the initial data, Proc. Am. Math. Soc., 102 (1988), 874-878. doi: 10.2307/2047326.

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