Energy methods for abstract nonlinear Schrödinger equations

Noboru Okazawa; Toshiyuki Suzuki; Tomomi Yokota

doi:10.3934/eect.2012.1.337

Article Contents

2012, Volume 1, Issue 2: 337-354. Doi: 10.3934/eect.2012.1.337

This issue Previous Article On the exponential stabilization of a thermo piezoelectric/piezomagnetic system Next Article Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise

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

Received: May 31, 2012

Revised: June 30, 2012

Published: November 2012

Abstract Related Papers Cited by

Abstract

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].

Keywords:

Energy methods,
abstract nonlinear Schrödinger equation,
nonnegative selfadjoint operators,
inverse-square potentials.

Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q15.

Citation:

Related Papers

Cited by

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.