The Mass-Critical for the Nonlinear Schrödinger Equation in *d* = 2

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.

Cite this paper

M. M-Ali, "The Mass-Critical for the Nonlinear Schrödinger Equation in*d* = 2," *Advances in Pure Mathematics*, Vol. 3 No. 5, 2013, pp. 482-490. doi: 10.4236/apm.2013.35070.

M. M-Ali, "The Mass-Critical for the Nonlinear Schrödinger Equation in

References

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[2] 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

[3] Y. Tsutsumi, “L2-Solutions for Nonlinear Schrodinger Equations and Nonlinear Groups,” Funkcialaj Ekvacioj, Vol. 30, No. 1, 1987, pp. 115-125.

[4] T. Tao, “Nonlinear Dispersive Equations: Local and Global Analysis,” CBMS Series, 106, American Mathematical Society, Providence, RI, 2006.

[5] 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

[6] 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

[7] J. Bourgain, “Global Solutions of Nonlinear Schrodinger Equations,” American Mathematical Society Colloquium Publications, Vol. 46, American Mathematical Society, Providence, RI, 1999.

[8] 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.

[9] 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.

[10] T. Cazenave, “An Introduction to Nonlinear Schrodinger Equations, First Edition,” Textos de Métodos Matemáticos 22, IM-UFRJ, Rio de Janeiro, 1989.

[11] 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.

[12] 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.

[13] J. Bourgain, “Refinements of Strichartz’ Inequality and Applications to 2D-NLS with Critical Nonlinearity,” International Mathematics Research Notices, No. 5, 1998, 253-283.

[14] T. Tao, M. Visan and X. Zhang, “Global Well-Posedness and Scattering for the Mass-Critical Nonlinear Schrodinger Equation for Radial Data in Higher Dimensions” Duke Mathematical Journal, Vol. 140, 2007, pp. 165-202.

[15] T. Tho and M. Visan, “Stability of Energy-Critical Nonlinear Schrodinger Equations in High Dimensions,” Electronic Journal of Differential Equations, Vol. 2005, No. 118, 2005, pp. 1-28.

[16] T. Tao, M. Visan and X. Zhang, “Global Well-Posedness and Scattering for the Defocusing Mass-Critical Nonlinear Schrodinger Equation for Radial Data in High Dimensions,” Duke Mathematical Journal, Vol. 140, 2007, No. 1, pp. 165-202.

[17] T. Tao, M. Visan and X. Zhang, “The Nonlinear Schrodinger Equation with Combined Power-Type Nonlinearities,” Communications in Partial Differential Equations, Vol. 32, No. 7-9, 2007, pp. 1281-1343. doi:10.1080/03605300701588805

[1] 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

[2] 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

[3] Y. Tsutsumi, “L2-Solutions for Nonlinear Schrodinger Equations and Nonlinear Groups,” Funkcialaj Ekvacioj, Vol. 30, No. 1, 1987, pp. 115-125.

[4] T. Tao, “Nonlinear Dispersive Equations: Local and Global Analysis,” CBMS Series, 106, American Mathematical Society, Providence, RI, 2006.

[5] 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

[6] 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

[7] J. Bourgain, “Global Solutions of Nonlinear Schrodinger Equations,” American Mathematical Society Colloquium Publications, Vol. 46, American Mathematical Society, Providence, RI, 1999.

[8] 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.

[9] 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.

[10] T. Cazenave, “An Introduction to Nonlinear Schrodinger Equations, First Edition,” Textos de Métodos Matemáticos 22, IM-UFRJ, Rio de Janeiro, 1989.

[11] 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.

[12] 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.

[13] J. Bourgain, “Refinements of Strichartz’ Inequality and Applications to 2D-NLS with Critical Nonlinearity,” International Mathematics Research Notices, No. 5, 1998, 253-283.

[14] T. Tao, M. Visan and X. Zhang, “Global Well-Posedness and Scattering for the Mass-Critical Nonlinear Schrodinger Equation for Radial Data in Higher Dimensions” Duke Mathematical Journal, Vol. 140, 2007, pp. 165-202.

[15] T. Tho and M. Visan, “Stability of Energy-Critical Nonlinear Schrodinger Equations in High Dimensions,” Electronic Journal of Differential Equations, Vol. 2005, No. 118, 2005, pp. 1-28.

[16] T. Tao, M. Visan and X. Zhang, “Global Well-Posedness and Scattering for the Defocusing Mass-Critical Nonlinear Schrodinger Equation for Radial Data in High Dimensions,” Duke Mathematical Journal, Vol. 140, 2007, No. 1, pp. 165-202.

[17] T. Tao, M. Visan and X. Zhang, “The Nonlinear Schrodinger Equation with Combined Power-Type Nonlinearities,” Communications in Partial Differential Equations, Vol. 32, No. 7-9, 2007, pp. 1281-1343. doi:10.1080/03605300701588805