Small data scattering of Hartree type fractional Schr\"odinger equations in dimension 2 and 3
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In this paper we study the small-data scattering of the $d$ dimensional fractional Schr\"odinger equations with $d = 2, 3$, L\'evy index $1 < \al < 2$ and Hartree type nonlinearity $F(u) = \mu(|x|^{-\gamma}*|u|^2)u$ with $ \max(\al, \frac{2d}{2d-1}) < \gamma \le 2$, $\gamma < d$. This equation is scaling-critical in $\dot H^{s_c}$, $s_c = \frac{\gamma-\al}2$. We show that the solution scatters in $H^{s,1}$ for any $s > s_c$, where $H^{s, 1}$ is a space of Sobolev type taking in angular regularity with norm defined by $\|\varphi\|_{H^{s, 1}} = \|\varphi\|_{H^s} + \|\nabla_{\mathbb S} \varphi\|_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere $\mathbb S^{d-1}$ and utilize $U^p$-$V^p$ space argument.
Abstract
In this paper we study the small-data scattering of the $d$ dimensional fractional Schr\"odinger equations with $d = 2, 3$, L\'evy index $1 < \al < 2$ and Hartree type nonlinearity $F(u) = \mu(|x|^{-\gamma}*|u|^2)u$ with $ \max(\al, \frac{2d}{2d-1}) < \gamma \le 2$, $\gamma < d$. This equation is scaling-critical in $\dot H^{s_c}$, $s_c = \frac{\gamma-\al}2$. We show that the solution scatters in $H^{s,1}$ for any $s > s_c$, where $H^{s, 1}$ is a space of Sobolev type taking in angular regularity with norm defined by $\|\varphi\|_{H^{s, 1}} = \|\varphi\|_{H^s} + \|\nabla_{\mathbb S} \varphi\|_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere $\mathbb S^{d-1}$ and utilize $U^p$-$V^p$ space argument.
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