On Fractional Schrödinger Equations in sobolev spaces

Younghun Hong (1) , Yannick Sire (2)

1 University of Texas at Austin [Austin]
2 I2M - Institut de Mathématiques de Marseille

Let $\sigma\in(0,1)$ with $\sigma\neq\frac{1}{2}$. We investigate the fractional nonlinear Schr\"odinger equation in $\mathbb R^d$: $$i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-\Delta)^\sigma$ is the Fourier multiplier of symbol $|\xi|^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

Mots clés: fractional Schrödinger; Sobolev spaces; local and global well-posedness; ill-posedness

Domaines: Equations aux dérivées partielles [math.AP]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (6), pp.2265--2282. ⟨10.3934/cpaa.2015.14.2265⟩. ⟨hal-01338426⟩