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

On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

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 $\alpha/r+n/w=\alpha$.
Citation: Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 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, (1976).

[2]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation,, Comm. Math. Phys., 147 (1992), 75. doi: 10.1007/BF02099529.

[3]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., 62 (2013), 991. doi: 10.1512/iumj.2013.62.4970.

[4]

M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. 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. 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. 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. doi: 10.1007/s00208-005-0672-0.

[8]

D. Foschi, Inhomogeneous Strichartz estimates,, J. Hyperbolic Differ. Equ., 2 (2005), 1. 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.

[10]

L. Grafakos, Classical Fourier Analysis,, $2^{nd}$ edition, (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. 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, 23 (1994), 223.

[13]

Y. Ke, Remark on the Strichartz estimates in the radial case,, J. Math. Anal. Appl., 387 (2012), 857. doi: 10.1016/j.jmaa.2011.09.039.

[14]

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

[15]

Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation,, J. Math. Anal. Appl., 373 (2011), 147. doi: 10.1016/j.jmaa.2010.06.019.

[16]

N. Laskin, Fractional quantum mechanics and Lévy path integrals,, Phys. Lett. A, 268 (2000), 298. 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. 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. 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. 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. 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. doi: 10.4171/RMI/591.

[22]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, Princeton Mathematical Series, (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. 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. doi: 10.1090/S0002-9947-06-04099-2.

show all references

References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction,, Springer-Verlag, (1976).

[2]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation,, Comm. Math. Phys., 147 (1992), 75. doi: 10.1007/BF02099529.

[3]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., 62 (2013), 991. doi: 10.1512/iumj.2013.62.4970.

[4]

M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. 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. 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. 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. doi: 10.1007/s00208-005-0672-0.

[8]

D. Foschi, Inhomogeneous Strichartz estimates,, J. Hyperbolic Differ. Equ., 2 (2005), 1. 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.

[10]

L. Grafakos, Classical Fourier Analysis,, $2^{nd}$ edition, (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. 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, 23 (1994), 223.

[13]

Y. Ke, Remark on the Strichartz estimates in the radial case,, J. Math. Anal. Appl., 387 (2012), 857. doi: 10.1016/j.jmaa.2011.09.039.

[14]

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

[15]

Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation,, J. Math. Anal. Appl., 373 (2011), 147. doi: 10.1016/j.jmaa.2010.06.019.

[16]

N. Laskin, Fractional quantum mechanics and Lévy path integrals,, Phys. Lett. A, 268 (2000), 298. 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. 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. 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. 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. 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. doi: 10.4171/RMI/591.

[22]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, Princeton Mathematical Series, (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. 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. doi: 10.1090/S0002-9947-06-04099-2.

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