December  2012, 1(2): 337-354. doi: 10.3934/eect.2012.1.337

Energy methods for abstract nonlinear Schrödinger equations

1. 

Department of Mathematics, Science University of Tokyo, 1-3 Kagurazaka, Sinjuku-ku, Tokyo 162-8601

2. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  June 2012 Revised  July 2012 Published  October 2012

So far there seems to be no abstract formulations for nonlinear Schrödinger equations (NLS). In some sense Cazenave[2, Chapter 3] has given a guiding principle to replace the free Schrödinger group with the approximate identity of resolvents. In fact, he succeeded in separating the existence theory from the Strichartz estimates. This paper is a proposal to extend his guiding principle by using the square root of the resolvent. More precisely, the abstract theory here unifies the local existence of weak solutions to (NLS) with not only typical nonlinearities but also some critical cases. Moreover, the theory yields the improvement of [21].
Citation: Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
References:
[1]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.

[2]

T. Cazenave, "Semilinear Schrödinger Equations,'' Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, 2003.

[3]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,'' Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.

[4]

T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494. doi: 10.1007/BF01258601.

[5]

T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), 18-29, Lecture Notes in Math., 1394, Springer, Berlin, 1989.

[6]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[7]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4.

[8]

J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 309-323.

[9]

M. J. Goldberg, L. Vega and N. Visciglia, Couterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math. Res. Not., (2006), Art. ID 13927, 16 pp.

[10]

H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.

[11]

R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation, Nonlinear Anal., 15 (1990), 479-495. doi: 10.1016/0362-546X(90)90128-4.

[12]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[13]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.

[14]

T. Kato, Nonlinear Schrödinger equations, in "Schrödinger Operators" (Eds. H. Holden and A. Jensen), in Lecture Notes in Physics, Springer, Berlin, 345 (1989), 218-263.

[15]

T. Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794.

[16]

Y. Maeda and N. Okazawa, Holomorphic families of Schrödinger operators in $L^p$, SUT J. Math., 47 (2011), 185-216.

[17]

T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. doi: 10.1016/0362-546X(90)90104-O.

[18]

M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct. Anal., 261 (2011), 90-110. doi: 10.1016/j.jfa.2011.03.010.

[19]

N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239.

[20]

N.\,Okazawa, Gauss hypergeometric functions of operators unifying fractional powers and logarithms, Semigroups of Operators: Theory and Applications (Rio de Janeiro, 2001), 209-219, Optimization Software, New York, 2002.

[21]

N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. doi: 10.1080/00036811.2011.631914).

[22]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, Grenoble, 45 (1995), 513-546. doi: 10.5802/aif.1463.

[23]

T. Suzuki, Energy methods for Hartree type equations with inverse-square potentials,, preprint., (). 

[24]

H. Tanabe, "Equations of Evolution,'' Monographs and Studies in Mathematics vol. 6, Pitman, London, 1979.

[25]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

[26]

Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal., 11 (1987), 1143-1154. doi: 10.1016/0362-546X(87)90003-4.

[27]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equation in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102. doi: 10.1512/iumj.1980.29.29007.

show all references

References:
[1]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.

[2]

T. Cazenave, "Semilinear Schrödinger Equations,'' Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, 2003.

[3]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,'' Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.

[4]

T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494. doi: 10.1007/BF01258601.

[5]

T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), 18-29, Lecture Notes in Math., 1394, Springer, Berlin, 1989.

[6]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[7]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4.

[8]

J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 309-323.

[9]

M. J. Goldberg, L. Vega and N. Visciglia, Couterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math. Res. Not., (2006), Art. ID 13927, 16 pp.

[10]

H. Hoshino and Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494.

[11]

R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation, Nonlinear Anal., 15 (1990), 479-495. doi: 10.1016/0362-546X(90)90128-4.

[12]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[13]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.

[14]

T. Kato, Nonlinear Schrödinger equations, in "Schrödinger Operators" (Eds. H. Holden and A. Jensen), in Lecture Notes in Physics, Springer, Berlin, 345 (1989), 218-263.

[15]

T. Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794.

[16]

Y. Maeda and N. Okazawa, Holomorphic families of Schrödinger operators in $L^p$, SUT J. Math., 47 (2011), 185-216.

[17]

T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal., 14 (1990), 765-769. doi: 10.1016/0362-546X(90)90104-O.

[18]

M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct. Anal., 261 (2011), 90-110. doi: 10.1016/j.jfa.2011.03.010.

[19]

N. Okazawa, $L^p$-theory of Schrödinger operators with strongly singular potentials, Japan. J. Math., 22 (1996), 199-239.

[20]

N.\,Okazawa, Gauss hypergeometric functions of operators unifying fractional powers and logarithms, Semigroups of Operators: Theory and Applications (Rio de Janeiro, 2001), 209-219, Optimization Software, New York, 2002.

[21]

N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. doi: 10.1080/00036811.2011.631914).

[22]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, Grenoble, 45 (1995), 513-546. doi: 10.5802/aif.1463.

[23]

T. Suzuki, Energy methods for Hartree type equations with inverse-square potentials,, preprint., (). 

[24]

H. Tanabe, "Equations of Evolution,'' Monographs and Studies in Mathematics vol. 6, Pitman, London, 1979.

[25]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

[26]

Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal., 11 (1987), 1143-1154. doi: 10.1016/0362-546X(87)90003-4.

[27]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equation in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102. doi: 10.1512/iumj.1980.29.29007.

[1]

Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations and Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531

[2]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations and Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022

[3]

Toshiyuki Suzuki. Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Conference Publications, 2015, 2015 (special) : 1019-1024. doi: 10.3934/proc.2015.1019

[4]

Toshiyuki Suzuki. Semilinear Schrödinger evolution equations with inverse-square and harmonic potentials via pseudo-conformal symmetry. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4347-4377. doi: 10.3934/cpaa.2021163

[5]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649

[6]

Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65

[7]

Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162

[8]

Vagif S. Guliyev, Ramin V. Guliyev, Mehriban N. Omarova, Maria Alessandra Ragusa. Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 671-690. doi: 10.3934/dcdsb.2019260

[9]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427

[10]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387

[11]

Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221

[12]

Hisashi Morioka. Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice. Inverse Problems and Imaging, 2011, 5 (3) : 715-730. doi: 10.3934/ipi.2011.5.715

[13]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623

[14]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[15]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control and Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011

[16]

Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024

[17]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

[18]

Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040

[19]

Xinlin Cao, Yi-Hsuan Lin, Hongyu Liu. Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators. Inverse Problems and Imaging, 2019, 13 (1) : 197-210. doi: 10.3934/ipi.2019011

[20]

Woocheol Choi, Yong-Cheol Kim. The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1993-2010. doi: 10.3934/cpaa.2018095

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (155)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]