Local well-posedness for quadratic nonlinear Schrödinger equations and the ``good'' Boussinesq equation

By Nobu Kishimoto, K. Tsugawa. Differential and Integral Equations, 2010

52 citations recorded by the source corpus.

. The Cauchy problem for 1-D nonlinear Schr¨odinger equations with quadratic nonlinearities are considered in the spaces H s,a defined by (cid:107) f (cid:107) H s,a = (cid:107) (1+ | ξ | ) s − a | ξ | a b f (cid:107) L 2 , and sharp local well-posedness and ill-posedness results are obtained in these spaces for nonlinearities including the term u ¯ u . In particular, when a = 0 the previous well-posedness result in H s , s > − 1 / 4, given by Kenig, Ponce and Vega (1996), is improved to s ≥ − 1 / 4. This also extends the result in H s,a by Otani (2004). The proof is based on an iteration argument similar to that of Kenig, Ponce and Vega, with a modification of the spaces of the Fourier restriction norm. Our result is also applied to the “good” Boussinesq equation and yields local well-posedness in H s × H s − 2 with s > − 1 / 2, which is an improvement of the previous result given by Farah (2009).

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91de651163b97ec7feece1b82f820c68d7d60b9d