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Generalized linear models for population dynamics in two juxtaposed habitats
Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ
Laboratoire Paul Painlevé (U.M.R. CNRS 8524), U.F.R. de Mathématiques, Université Lille 1, 59655 Villeneuve d'Ascq Cedex, France |
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 [
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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