Research ArticleOpen AccessGoogle Scholar indexed
The Mass-Critical for the Nonlinear Schrödinger Equation in <i>d</i> = 2
Department of Mathematics, Faculty of Education, Kassala University, Kassala, Sudan
- 1 Department of Mathematics, Faculty of Education, Kassala University, Kassala, Sudan
Advances in Pure Mathematics·Volume 03 (2013)·Pages 482–490·Published 23 July 2013·DOI10.4236/apm.2013.35070
Copy link · social · email
Abstract
This paper studies the global behavior defocusing nonlinear Schrodinger equation in dimension d = 2, and we will discuss the case . This means that the solutions , and called critical solution. We show that u scatters forward and backward to a free solution and the solution is globally well posed.
KeywordsNLSWell Posed
- T. Cazenave and F. Weissler, “The Cauchy Problem for the Nonlinear Schrodinger Equation in H1,” Manuscripta Mathematics, Vol. 61, No. 4, 1988, pp. 477-494. doi:10.1007/BF01258601
- T. Cazenave and F. Weissler, “The Cauchy Problem for the Nonlinear Schrodinger Equation in Hs,” Nonlinear Analysis, Vol. 14, No. 10, 1990, pp. 807-836. doi:10.1016/0362-546X(90)90023-A
- Y. Tsutsumi, “L2-Solutions for Nonlinear Schrodinger Equations and Nonlinear Groups,” Funkcialaj Ekvacioj, Vol. 30, No. 1, 1987, pp. 115-125.
- T. Tao, “Nonlinear Dispersive Equations: Local and Global Analysis,” CBMS Series, 106, American Mathematical Society, Providence, RI, 2006.
- T. Tao, “Spherically Averaged Endpoint Stnchartz Estimates for the Two-Dimensional Schrodinger Equation,” Communications in Partial Differential Equations, Vol. 25, No. 7-8, 2000, pp. 1471-1485. doi:10.1080/03605300008821556
- T. Tao, M. Visan and X. Zhang, “Minimal-Mass Blow up Solutions of the Mass-Critical NLS,” Forum Mathematicum, Vol. 20, No. 5, 2008, pp. 881-919. doi:10.1515/FORUM.2008.042
- J. Bourgain, “Global Solutions of Nonlinear Schrodinger Equations,” American Mathematical Society Colloquium Publications, Vol. 46, American Mathematical Society, Providence, RI, 1999.
- J. Dolbeault and G. Rein, “Time-Dependent Rescalings and Lyapunov Functionals for the Vlasov-Poisson Ena d Euler-Poisson Systems, and for Related Models of Kinetic Equations, fluid Dynamics and Quantum Mechanics,” Mathematical Models and Methods in Applied Sciences, Vol. 11, 2001, pp. 407-432.
- J. Colliander, M. Grillakis and N. Tzirakis, “Improved Interaction Morawetz Inequalities for the Cubic Nonlinear Schrodinger Equation on R2,” International Mathematics Research Notices, No. 23, 2007, pp. 90-119.
- T. Cazenave, “An Introduction to Nonlinear Schrodinger Equations, First Edition,” Textos de Métodos Matemáticos 22, IM-UFRJ, Rio de Janeiro, 1989.
- M. Del Pino and J. Dolbeault, “Best Constants for Gagliardo-Nirenberg Inequalities and Applications to Nonlinear Diffusions,” Journal de Mathématiques Pures et Appliquées, Vol. 81, No. 9, 2002, pp. 847-875.
- M. Del Pino and J. Dolbeault, “Generalized Sobolev Ine-Qualities and Asymptotic Behaviour in Fast Diffusion and Porous Media Problems,” Ceremade, 9905, 1999, pp. 1-45.
- J. Bourgain, “Refinements of Strichartz’ Inequality and Applications to 2D-NLS with Critical Nonlinearity,” International Mathematics Research Notices, No. 5, 1998, 253-283.