Dispersive estimates and the 2D cubic NLS

Fabrice Planchon

Description

Dispersive estimates and the 2D cubic NLS equation Fabrice Planchon Abstract We prove that the initial value problem for the 2D cubic semi-linear Schrodinger equation is well-posed in the Besov space _ B 0;1 2 (R 2 ). For this, we rely on some new dispersive inequalities derived from bilinear restriction theorems. Introduction We are interested in the Cauchy problem for the NLS equation (1) i@ t u+ u = u 3 ; u(x; 0) = u 0 (x) in R 2 . Since we will be dealing with small data, the sign of the non-linearity is irrelevant, and u 3 should be understood as being any cubic combination of u and u. Recall (1) is locally well-posed ( and globally for small data) in L 2 ([4]), and L 2 is the scale-invariant space: if u is a solution, then kuk L 2 = ku k L 2 , where (2) u 0 (x) ! u 0; (x) = u 0 (x) u(x; t) ! u (x; t) = u(x; 2 t): Indeed, this

• pseudo-conformal inversion


• space lebesgue spaces


• solutions provided


• space


• based besov


• lebesgue space-time

Sujets

• Planchon

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

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