On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

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April 2016, 36(4): 1905-1926. doi: 10.3934/dcds.2016.36.1905

On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

Chu-Hee Cho ^1, , Youngwoo Koh ^2, and Ihyeok Seo ^3,

1.	Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea
2.	School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, South Korea
3.	Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea

Received January 2015 Revised July 2015 Published September 2015

Abstract
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In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrödinger equation in the radial case. Then we apply them to the well-posedness theory for the equation $i\partial_{t}u+|\nabla|^{\alpha}u=V(x,t)u$, $1<\alpha<2$, with radial $\dot{H}^\gamma$ initial data below $L^2$ and radial potentials $V\in L_t^rL_x^w$ under the scaling-critical range $\alpha/r+n/w=\alpha$.

Keywords: Strichartz estimates, well-posedness, Schrödinger equations..

Mathematics Subject Classification: Primary: 35B45, 35A01; Secondary: 35Q4.

Citation: Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905

References:

[1]	J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976.
[2]	T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529.
[3]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970.
[4]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.
[5]	E. Cordero and F. Nicola, Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation, Math. Nachr., 281 (2008), 25-41. doi: 10.1002/mana.200610585.
[6]	E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential equations., 245 (2008), 1945-1974. doi: 10.1016/j.jde.2008.07.009.
[7]	P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schrödinger equation with a potential in $L_t^rL_x^s$, Math. Ann., 333 (2005), 271-290. doi: 10.1007/s00208-005-0672-0.
[8]	D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.
[9]	J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéare, 2 (1985), 309-327.
[10]	L. Grafakos, Classical Fourier Analysis, $2^{nd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2008.
[11]	Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.
[12]	T. Kato, An $L^{q,r}$ -theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 223-238.
[13]	Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.
[14]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[15]	Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160. doi: 10.1016/j.jmaa.2010.06.019.
[16]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.
[17]	S. Lee and I. Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl., 389 (2012), 461-468. doi: 10.1016/j.jmaa.2011.11.067.
[18]	S. Lee and I. Seo, On inhomogeneous Strichartz estimates for the Schrödinger equation, Rev. Mat. Iberoam., 30 (2014), 711-726. doi: 10.4171/RMI/797.
[19]	V. Naibo and A. Stefanov, On some Schrödinger and wave equations with time dependent potentials, Math. Ann., 334 (2006), 325-338. doi: 10.1007/s00208-005-0720-9.
[20]	I. Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J., 60 (2011), 1203-1227. doi: 10.1512/iumj.2011.60.4824.
[21]	S. Shao, Sharp linear and bilinear restriction estimate for paraboloids in the cylinderically symmetric case, Rev. Mat. Iberoam., 25 (2009), 1127-1168. doi: 10.4171/RMI/591.
[22]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. 1993.
[23]	R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.
[24]	M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2.

show all references

References:

[1]	J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976.
[2]	T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529.
[3]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970.
[4]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.
[5]	E. Cordero and F. Nicola, Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation, Math. Nachr., 281 (2008), 25-41. doi: 10.1002/mana.200610585.
[6]	E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential equations., 245 (2008), 1945-1974. doi: 10.1016/j.jde.2008.07.009.
[7]	P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schrödinger equation with a potential in $L_t^rL_x^s$, Math. Ann., 333 (2005), 271-290. doi: 10.1007/s00208-005-0672-0.
[8]	D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.
[9]	J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéare, 2 (1985), 309-327.
[10]	L. Grafakos, Classical Fourier Analysis, $2^{nd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2008.
[11]	Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.
[12]	T. Kato, An $L^{q,r}$ -theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 223-238.
[13]	Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.
[14]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[15]	Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160. doi: 10.1016/j.jmaa.2010.06.019.
[16]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.
[17]	S. Lee and I. Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl., 389 (2012), 461-468. doi: 10.1016/j.jmaa.2011.11.067.
[18]	S. Lee and I. Seo, On inhomogeneous Strichartz estimates for the Schrödinger equation, Rev. Mat. Iberoam., 30 (2014), 711-726. doi: 10.4171/RMI/797.
[19]	V. Naibo and A. Stefanov, On some Schrödinger and wave equations with time dependent potentials, Math. Ann., 334 (2006), 325-338. doi: 10.1007/s00208-005-0720-9.
[20]	I. Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J., 60 (2011), 1203-1227. doi: 10.1512/iumj.2011.60.4824.
[21]	S. Shao, Sharp linear and bilinear restriction estimate for paraboloids in the cylinderically symmetric case, Rev. Mat. Iberoam., 25 (2009), 1127-1168. doi: 10.4171/RMI/591.
[22]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. 1993.
[23]	R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.
[24]	M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2.

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