On quadratic Schrdinger equations in R 11 : a normal form approach

ISSN： 0024-6107

Source： Journal of the London Mathematical Society, Vol.86, Iss.2, 2012-10, pp. : 499-519

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Abstract

For the Schrdinger equation u_tiu_xx [u²], (0, ), we establish local well-posedness in H¹ (note that if 0, this matches, up to an endpoint, the sharp result of BejenaruTao [Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrdinger equation, J. Funct. Anal. 233 (2006) 228259.]). Our approach differs significantly from the previous one, we use normal form transformation to analyze the worst interacting terms in the nonlinearity and then show that the remaining terms are (much) smoother. In particular, this allows us to conclude that ueit_x² u(0) H^1/2 (R¹), even though u(0) H¹.In addition, as a byproduct of our normal form analysis, we obtain a Lipschitz continuity property in H^1/2 of the solution operator (which originally acts on H¹), which is new even in the case 0. As an easy corollary, we obtain local well-posedness results for u_t iu_xxzz.Finally, we sketch an approach to obtain similar results for the equations u_t iu_xx[u] and u_t iu_xx[²].