American Institute of Mathematical Sciences

Advanced
March  2014, 3(1): 15-33. doi: 10.3934/eect.2014.3.15

Well-posedness for degenerate Schrödinger equations

Massimo Cicognani 1, and  Michael Reissig 2,

1. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna, Italy
2. 

TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Institut für Angewandte Analysis, 09596 Freiberg, Germany

Received  March 2013 Revised  August 2013 Published  February 2014

We consider the initial value problem for Schrödinger type equations $$\frac{1}{i}\partial_tu-a(t)\Delta_xu+\sum_{j=1}^nb_j(t,x)\partial_{x_j}u=0$$ with $a(t)$ vanishing of finite order at $t=0$ proving the well-posedness in Sobolev and Gevrey spaces according to the behavior of the real parts $\Re b_j(t,x)$ as $t\to0$ and $|x|\to\infty$. Moreover, we discuss the application of our approach to the case of a general degeneracy.
Keywords: Gevrey well-posedness., Schrödinger equations, Cauchy problem, time-depending Hamiltonian, Levi conditions.
Mathematics Subject Classification: Primary: 35J10; Secondary: 35Q4.
Citation: Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
References:
[1]

A. Ascanelli, M. Cicognani and F. Colombini, The global Cauchy problem for a vibrating beam equation,, Journal of Differential Equations, 247 (2009), 1440.  doi: 10.1016/j.jde.2009.06.012.  Google Scholar

[2]

A. Ascanelli and M. Cicognani, Gevrey solutions for a vibrating beam equation,, Rend. Semin. Mat. Torino, 67 (2009), 151.   Google Scholar

[3]

M. Cicognani and F. Colombini, Optimal well-posedness of the cauchy problem for evolution equations with $C^N$ coefficients,, Differential and Integral Equations, 17 (2004), 1079.   Google Scholar

[4]

M. Cicognani and T. Herrmanni, $H^\infty$ well-posedness for a $2$-evolution Cauchy problem with complex coefficients,, Journal of Pseudo-Differential Operators and Applications, 4 (2013), 63.  doi: 10.1007/s11868-013-0062-4.  Google Scholar

[5]

S. I. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions,, J. Math. Kyoto Univ., 34 (1994), 319.   Google Scholar

[6]

S. I. Doi, Remarks on the Cauchy problem for Schrödingerr-type equations,, Comm. Partial Differential Equations, 21 (1996), 163.  doi: 10.1080/03605309608821178.  Google Scholar

[7]

M. Dreher, Necessary conditions for the well-posedness of Schrödinger type equations in Gevrey spaces,, Bull. Sci. Math., 127 (2003), 485.  doi: 10.1016/S0007-4497(03)00026-5.  Google Scholar

[8]

W. Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations,, Osaka J. Math., 21 (1984), 565.   Google Scholar

[9]

W. Ichinose, Sufficient condition on $H^\infty$ well-posedness for Schrödinger type equations,, Comm. Partial Differential Equations, 9 (1984), 33.  doi: 10.1080/03605308408820324.  Google Scholar

[10]

W. Ichinose, On a necessary condition for $L^2$ well-posedness of the Cauchy problem for some Schrödinger type equations with a potential term,, J. Math. Kyoto Univ., 33 (1993), 647.   Google Scholar

[11]

W. Ichinose, On the Cauchy problem for Schrödinger type equations and Fourier integral operators,, J. Math. Kyoto Univ., 33 (1993), 583.   Google Scholar

[12]

K. Kajitani, The Cauchy problem for Schrödinger type equations with variable coefficients,, J. Math. Soc. Japan, 50 (1998), 179.  doi: 10.2969/jmsj/05010179.  Google Scholar

[13]

K. Kajitani and A. Baba, The Cauchy problem for Schrödinger type equations,, Bull. Sci. Math., 119 (1995), 459.   Google Scholar

[14]

K. Kajitani and T. Nishitani, The Hyperbolic Cauchy Problem,, Lecture Notes in Mathematics, (1991).   Google Scholar

[15]

S. Mizohata, On some Schrödinger type equations,, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981), 81.  doi: 10.3792/pjaa.57.81.  Google Scholar

show all references

References:
[1]

A. Ascanelli, M. Cicognani and F. Colombini, The global Cauchy problem for a vibrating beam equation,, Journal of Differential Equations, 247 (2009), 1440.  doi: 10.1016/j.jde.2009.06.012.  Google Scholar

[2]

A. Ascanelli and M. Cicognani, Gevrey solutions for a vibrating beam equation,, Rend. Semin. Mat. Torino, 67 (2009), 151.   Google Scholar

[3]

M. Cicognani and F. Colombini, Optimal well-posedness of the cauchy problem for evolution equations with $C^N$ coefficients,, Differential and Integral Equations, 17 (2004), 1079.   Google Scholar

[4]

M. Cicognani and T. Herrmanni, $H^\infty$ well-posedness for a $2$-evolution Cauchy problem with complex coefficients,, Journal of Pseudo-Differential Operators and Applications, 4 (2013), 63.  doi: 10.1007/s11868-013-0062-4.  Google Scholar

[5]

S. I. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions,, J. Math. Kyoto Univ., 34 (1994), 319.   Google Scholar

[6]

S. I. Doi, Remarks on the Cauchy problem for Schrödingerr-type equations,, Comm. Partial Differential Equations, 21 (1996), 163.  doi: 10.1080/03605309608821178.  Google Scholar

[7]

M. Dreher, Necessary conditions for the well-posedness of Schrödinger type equations in Gevrey spaces,, Bull. Sci. Math., 127 (2003), 485.  doi: 10.1016/S0007-4497(03)00026-5.  Google Scholar

[8]

W. Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations,, Osaka J. Math., 21 (1984), 565.   Google Scholar

[9]

W. Ichinose, Sufficient condition on $H^\infty$ well-posedness for Schrödinger type equations,, Comm. Partial Differential Equations, 9 (1984), 33.  doi: 10.1080/03605308408820324.  Google Scholar

[10]

W. Ichinose, On a necessary condition for $L^2$ well-posedness of the Cauchy problem for some Schrödinger type equations with a potential term,, J. Math. Kyoto Univ., 33 (1993), 647.   Google Scholar

[11]

W. Ichinose, On the Cauchy problem for Schrödinger type equations and Fourier integral operators,, J. Math. Kyoto Univ., 33 (1993), 583.   Google Scholar

[12]

K. Kajitani, The Cauchy problem for Schrödinger type equations with variable coefficients,, J. Math. Soc. Japan, 50 (1998), 179.  doi: 10.2969/jmsj/05010179.  Google Scholar

[13]

K. Kajitani and A. Baba, The Cauchy problem for Schrödinger type equations,, Bull. Sci. Math., 119 (1995), 459.   Google Scholar

[14]

K. Kajitani and T. Nishitani, The Hyperbolic Cauchy Problem,, Lecture Notes in Mathematics, (1991).   Google Scholar

[15]

S. Mizohata, On some Schrödinger type equations,, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981), 81.  doi: 10.3792/pjaa.57.81.  Google Scholar

[1]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[2]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
[3]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[4]

Fatih Bayazit, Britta Dorn, Marjeta Kramar Fijavž. Asymptotic periodicity of flows in time-depending networks. Networks & Heterogeneous Media, 2013, 8 (4) : 843-855. doi: 10.3934/nhm.2013.8.843
[5]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703
[6]

Shubin Wang, Guowang Chen. Cauchy problem for the nonlinear Schrödinger-IMBq equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 203-214. doi: 10.3934/dcdsb.2006.6.203
[7]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[8]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[9]

Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131
[10]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[11]

Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations & Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57
[12]

Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997
[13]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[14]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273
[15]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389
[16]

Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066
[17]

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144
[18]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[19]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[20]

Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences