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Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation
Authors:Ioan Bejenaru  Daniela De Silva
Institution:Department of Mathematics, University of California, Los Angeles, California 90095 ; Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
Abstract:We establish that the initial value problem for the quadratic non-linear Schrödinger equation

$\displaystyle iu_t - \Delta u = u^2,$

where $ u: \mathbb{R}^2 \times \mathbb{R} \to \mathbb{C}$, is locally well-posed in $ H^s(\mathbb{R}^2)$ when $ s > -1$. The critical exponent for this problem is $ s_c=-1$, and previous work by Colliander, Delort, Kenig and Staffilani, 2001, established local well-posedness for $ s > -3/4$.

Keywords:
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