Dispersive estimates and the 2D cubic NLS

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Fabrice Planchon

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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

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