Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation

Bejenaru, Ioan; De Silva, Daniela

doi:10.1090/S0002-9947-08-04415-2

Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation
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by Ioan Bejenaru and Daniela De Silva PDF

Trans. Amer. Math. Soc. 360 (2008), 5805-5830 Request permission

Abstract:

We establish that the initial value problem for the quadratic non-linear Schrödinger equation \[ iu_t - \Delta u = u^2,\] 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 $s > -3/4$.

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