Strichartz Estimates for Schrödinger Equations with Non-degenerate Coefficients*

Abstract

In the present paper, the full range Strichartz estimates for homogeneous Schrödinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained.

Keywords

Schrödinger equation / Strichartz estimates / Parametrix / 35Q40 / 35S30

References

[1] Burq Amer. J. Math., 2004, 126: 569

[2] Cazenave, T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, Vol. 10, A. M. S., Providence, RI, 2003

[3] Cazenave Nonlin. Anal., 1990, 14: 807

[4] Chihara Math. Japan, 1995, 42: 35

[5] Christ J. Func. Anal., 2001, 179: 409

[6] Craig Comm. Pure Appl. Math., 1995, 48: 769

[7] Doi J. Math. Kyoto Univ., 1994, 34: 319

[8] Doi Comm. Partial Differential Equations, 1996, 21: 163

[9] Ginibre J. Funct. Anal., 1995, 133: 50

[10] Hassel, A., Tao, T. and Wunsch, J., Sharp Strichartz estimates on non trapping asymptotically conic manifolds, preprint

[11] Hörmander, L., The analysis of linear partial differential operators, Vol. III, Grundlehren der Mathematischen Wissenschaften, 274, Springer, Berlin, 1985

[12] Ishimori Progr. Theor. Phys., 1984, 72: 33

[13] Kenig Adv. Math., 2005, 196: 373

[14] Keel Amer. J. Math., 1998, 120: 955

[15] Koch Comm. Pure Appl. Math., 2005, 58: 217

[16] Miao, C. X., Harmonic Analysis and Its Applications in Partial Differential Equations (in Chinese), Science Press, Beijing, 2000

[17] Planchon, F., Lectures on Schrödinger equation, COE Lecture Notes, 2003

[18] Robbiano, L. and Zuily, C., Strichartz estimates for Schrödinger equations with variable coefficients, preprint, 2005

[19] Salort Int. Math. Res. Not., 2005, 11: 687

[20] Smith Comm. Partial Differential Equations, 2000, 25: 2171

[21] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Monographs in Harmonic Analysis, Vol. III, Princeton University Press, Princeton, NJ, 1993

[22] Strichartz Duke Math. J., 1977, 44: 705

[23] Staffilani Comm. Partial Differential Equations, 2002, 27: 1337

[24] Sulem, C. and Sulem, P. L., The Nonlinear Schrödinger Equation. Self-focusing andWave Collapse, Applied Mathematical Sciences, Vol. 139, Springer-Verlag, New York, 1999

[25] Taylor, M. E., Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, New York, 1997

[26] Tataru Amer. J. Math., 2000, 122: 349

[27] Tataru Amer. J. Math., 2001, 123: 385

[28] Tataru J. Amer. Math. Soc., 2002, 15: 419

[29] Yajima Comm. Math. Phys., 1987, 110: 415

[30] Zakharov Physica D, 1980, 1: 192