Local Well-Posedness of the Periodic Nonlinear Schrödinger Equation with a Quadratic Nonlinearity u ¯ 2 in Negative Sobolev Spaces.

Abstract

We study low regularity local well-posedness of the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity  ${\overline{u}}^2$ , posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with respect to the $X^{s,b}$ -space is known to fail when the regularity s is below some threshold value, we establish local well-posedness for such low regularity by introducing modifications on the $X^{s,b}$ -space.