Well-posedeness for the non-isotropic Schrödinger equations on cylinders and periodic domains

Abstract

The initial value problem (IVP) for the non-isotropic Schrödinger equation posed on the two-dimensional cylinders and ${\mathbb{T}}^2$ is considered. The IVP is shown to be locally well-posed for small initial data in $H^s(\mathbb{T}\times \mathbb{R})$ if $s\ge 0$. For the IVP posed on $\mathbb{R}\times \mathbb{T}$, given data are considered in the anisotropic Sobolev spaces thereby obtaining the local well-posedness result in $H^{s_1,s_2}(\mathbb{R}\times \mathbb{T})$, if $s_1\ge 0$ and $s_2>\frac{1}{2}$. In the purely periodic case, a particular case of the IVP is shown to be locally well-posed for any given initial data in $H^s({\mathbb{T}}^2)$ if $s>\frac{1}{4}$. In some cases, ill-posedness issues are also considered showing that the IVP posed on $\mathbb{T}\times \mathbb{R}$, in the focusing case, is ill-posed in the sense that the application data-solution fails to be uniformly continuous for data in $H^s(\mathbb{T}\times \mathbb{R})$ if $-\frac{1}{2}\le s<0$.

24 pages