The cubic fourth-order Schrödinger equation

Journal of Functional Analysis

Volumn 256, Issue 8, 2009, Pages 2473-2517

  Pausader, Benoit ^a  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

Energy critical equation; Fourth order dispersive equation; Scattering

Indexed keywords

EID: 62049085138     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2008.11.009     Document Type: Article

References (40)

1. T. Alazard, R. Carles, Loss of regularity for supercritical nonlinear Schrödinger equations, Math. Ann., in press
2. Bahouri H., and Gerard P. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121 (1999) 131-175
3. Ben-Artzi M., Koch H., and Saut J.C. Dispersion estimates for fourth order Schrödinger equations. C. R. Math. Acad. Sci. Sér. 1 330 (2000) 87-92
4. Carles R. Geometric optics and instability for semi-classical Schrödinger equations. Arch. Ration. Mech. Anal. 183 3 (2007) 525-553
5. Cazenave T. Semilinear Schrödinger Equations. Courant Lect. Notes Math. vol. 10 (2003), New York University, Courant Institute of Mathematical Sciences, Amer. Math. Soc., New York, Providence, RI
6. M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Ann. Inst. H. Poincaré, in press
7. 3. Ann. of Math. (2) 167 3 (2008) 767-865
8. Fibich G., Ilan B., and Papanicolaou G. Self-focusing with fourth order dispersion. SIAM J. Appl. Math. 62 4 (2002) 1437-1462
9. Fibich G., Ilan B., and Schochet S. Critical exponent and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion. Nonlinearity 16 (2003) 1809-1821
10. Gerard P., Meyer Y., and Oru F. Inégalités de Sobolev précisées. Sémin. Equ. Deriv. Partielles (1996-1997), École polytechnique 11 pp
11. s. Differential Integral Equations 15 9 (2002) 1073-1083
12. Hao C., Hsiao L., and Wang B. Well-posedness for the fourth-order Schrödinger equations. J. Math. Anal. Appl. 320 (2006) 246-265
13. Hao C., Hsiao L., and Wang B. Well-posedness of the Cauchy problem for the fourth-order Schrödinger equations in high dimensions. J. Math. Anal. Appl. 328 (2007) 58-83
14. Huo Z., and Jia Y. The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament. J. Differential Equations 214 (2005) 1-35
15. Karpman V.I. Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations. Phys. Rev. E 53 2 (1996) 1336-1339
16. Karpman V.I., and Shagalov A.G. Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion. Phys. D 144 (2000) 194-210
17. s-solutions and unconditional well-posedness. J. Anal. Math. 67 (1995) 281-306
18. Kenig C., and Merle F. Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case. Invent. Math. 166 3 (2006) 645-675
19. C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing nonlinear wave equation, Acta Math., in press
20. Kenig C., Ponce G., and Vega L. Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40 (1991) 33-69
21. Keraani S. On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differential Equations 175 (2001) 353-392
22. R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint
23. R. Killip, T. Tao, M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, preprint
24. Lebeau G. Nonlinear optic and supercritical wave equation. Bull. Soc. Roy. Sci. Liege 70 4-6 (2001) 267-306 Hommage a Pascal Laubin, 2002
25. Lebeau G. Perte de régularité pour des équations d'onde sur-critiques. Bull. Soc. Math. France 133 1 (2005) 145-157
26. Lin J.E., and Strauss W.A. Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Funct. Anal. 30 (1978) 245-263
27. C. Miao, G. Xu, L. Zhao, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth order in dimensions d ≥ 9, preprint
28. Pausader B. Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ. 4 3 (2007) 197-225
29. B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst. Ser. A, in press
30. B. Pausader, Minimal mass blow-up solutions for the mass-critical fourth-order Schrödinger equation, preprint
31. B. Pausader, W.A. Strauss, Analyticity of the scattering operator for fourth-order nonlinear waves, preprint
32. 1 + 4. Amer. J. Math. 129 (2007) 1-60
33. Segata J. Well-posedness for the fourth-order nonlinear Schrödinger type equation related to the vortex filament. Differential Integral Equations 16 7 (2003) 841-864
34. Segata J. Remark on well-posedness for the fourth order nonlinear Schrödinger type equation. Proc. Amer. Math. Soc. 132 (2004) 3559-3568
35. Segata J. Modified wave operators for the fourth-order non-linear Schrödinger-type equation with cubic non-linearity. Math. Methods Appl. Sci. 26 15 (2006) 1785-1800
36. Tao T. Nonlinear Dispersive Equations, Local and Global Analysis. CBMS Reg. Conf. Ser. Math. vol. 106 (2006), Amer. Math. Soc., Providence, RI. 0-8218-4143-2 Published for the Conference Board of the Mathematical Science, Washington, DC
37. T. Tao, M. Visan, X. Zhang, Minimal-mass blow up solutions of the mass critical NLS, Forum Math., in press
38. Thomann L. Geometric and projective instability for the Gross-Pitaevski equation. Asymptot. Anal. 51 3-4 (2007) 271-287
39. Thomann L. Instabilities for supercritical Schrödinger equations in analytic manifolds. J. Differential Equations 245 1 (2008) 249-280
40. Visan M. The defocusing energy-critical nonlinear Schrodinger equation in higher dimensions. Duke Math. J. 138 (2007) 281-374