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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.
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참고문헌 (20)
참고문헌 신청
Y. Cho / 2017 / Short-range scattering of Hartree type fractional NLS / J. Differential Equations 262(1):116~144
Y. Cho / 2013 / On the Cauchy problem of fractional Schrodinger equation with Hartree type nonlinearity / Funkacial. Ekvac. 56(2):193~224
Y. Cho / 2016 / On small data scattering of Hartree equations with short-range interaction / Commun. Pure Appl. Anal. 15(5):1809~1823
Y. Cho / 2017 / Corrigendum to "On small data scattering of Hartree equations with shortrange interaction" [Comm. Pure Appl. Anal. 15 (5) (2016), 809-1823] / Commun. Pure Appl. Anal. 16(5):1939~1940
Y. Cho / On the modified scattering of 3-d Hartree type fractional Schrodinger equations with Coulomb potential
Y. Cho / 2013 / On the Cauchy problem of fractional Schrodinger equation with Hartree type nonlinearity / Funkacial. Ekvac. 56(2):193~224
Y. Cho / 2016 / On small data scattering of Hartree equations with short-range interaction / Commun. Pure Appl. Anal. 15(5):1809~1823
Y. Cho / 2017 / Corrigendum to "On small data scattering of Hartree equations with shortrange interaction" [Comm. Pure Appl. Anal. 15 (5) (2016), 809-1823] / Commun. Pure Appl. Anal. 16(5):1939~1940
Y. Cho / On the modified scattering of 3-d Hartree type fractional Schrodinger equations with Coulomb potential
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