April  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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[8]

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

[10]

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

[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.  Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  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.  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.  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.  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.  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.  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.  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.  doi: 10.4171/RMI/591.  Google Scholar

[22]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, Princeton Mathematical Series, (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.  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.  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, (1976).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[8]

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

[10]

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

[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.  Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  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.  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.  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.  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.  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.  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.  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.  doi: 10.4171/RMI/591.  Google Scholar

[22]

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

[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[3]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[4]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[5]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[6]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[7]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[8]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[9]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[10]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[11]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[14]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[15]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[16]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[17]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[18]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[19]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[20]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]