Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations

Abstract

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^s$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$.

26 pages

References in corpus (2)

• Convolutions of singular measures and applications to the Zakharov system
• On the 2d Zakharov system with L^2 Schrödinger data

Cited by in corpus (5)

• On the nonlinear Brascamp-Lieb inequality
• Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces
• Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with radial initial data
• Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations
• Grand Lebesgue Spaces norm estimates for multivariate functional operations