Local well-posedness for quadratic Schrödinger equations in $\mathbf{R^{1+1}}$: a normal form approach

Details Local well-posedness for quadratic Schrödinger equations in $\mathbf{R^{1+1}}$: a normal form approach Local well-posedness for quadratic Schrödinger equations in $\mathbf{R^{1+1}}$: a normal form approach

2011-05-09

arxiv.org/abs/1105.1713v1

Mathematics

Analysis of PDEs

Scientific paper

For the Schr\"odinger equation $u_t+i u_{xx}=\nab^\be[u^2]$, $\be\in (0,1/2)$, we establish local well-posedness in $H^{\be-1+}$ (note that if $\be=0$, this matches, up to an endpoint, the sharp result of Bejenaru-Tao, \cite{BT}). Our approach differs significantly from the previous ones in that 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 $u-e^{-i t \p_x^2} u(0)\in H^{-\f{1}{2}}(\rone)$, even though $u(0)\in H^{\be-1+}$. In addition and as a byproduct of our normal form analysis, we obtain a Lipschitz continuity property in $H^{-\f{1}{2}}$ of the solution operator (which originally acts on $H^{\be-1+}$), which is new even in the case $\be=0$. As an easy corollary, we obtain local well-posedness results for $u_t+ i u_{xx} = [\nab^{\beta} u]^2$. Finally, we sketch an approach to obtain similar statements for the equations $u_t+i u_{xx}=\nab^\be[u\bar{u}]$ and $u_t+i u_{xx}=\nab^\be[\bar{u}^2]$.

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