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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.  Google Scholar

[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.  Google Scholar

[10]

L. Grafakos, Classical Fourier Analysis, $2^{nd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.  Google Scholar

[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.  Google Scholar

[10]

L. Grafakos, Classical Fourier Analysis, $2^{nd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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