On fractional Schrödinger equations in Sobolev spaces
This paper considers the associated Cauchy problem for the following fractional nonlinear Schrödinger equation \[ i\partial_tu+(-\Delta)^{\sigma}u+\mu |u|^{p-1}u=0, \] where the fractional Laplacian \((-\Delta)^{\sigma}\) is a Fourier multiplier of symbol \(\left| \xi\right|^{2\sigma}\), \(\sigma\in (0,1)\), \(\sigma\neq 1/2\), and \(\mu= \pm 1\). The authors prove that the Cauchy problem is locally well-posed in \(H^s\) for \(s\geq \max(s_c,s_g,0)\) and it is ill-posed in \(H^s\) for \(s\in (s_c,0)\). Here the numbers \(s_c\) and \(s_g\) are defined by \[ s_c=\frac{N}{2}-\frac{2\sigma}{p-1}\;\;\mathrm{and}\;\;s_g=\frac{1-\sigma}{2}. \] The proof rely on the standard Strichartz estimates and functional inequalities in \(\mathbb{R}^N\).
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