论文标题
Sobolev-Lorentz空间，并应用于不均匀的Biharmonic NLS方程
Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation
论文作者
论文摘要
我们考虑了不均匀的双键式非固定schrödinger（ibnls）方程的库奇问题\ [iu_ {t} +δ^{2} u =λ| x | x | x |^{ - b} - b} - b} | r^{d}），\]其中$λ\ in \ mathbb r $，$ d \ in \ mathbb n $，$ 0 \ le s <\ s <\ min \ left \ left \ {2+ \ frac {d} {d} {2} {2} {2} {2}，d \ right \} d-s，\; 2+ \ frac {d} {2} -s \ right \} $和$ 0 <σ\ leσ_{c} $，带有$σ<\ infty $。这里$σ_{c}（s）= \ frac {8-2b} {d-2s} $如果$ s <\ frac {d} {2} {2} $和$σ_{c}（c}（s）= \ infty $ s = \ infty $如果$ s \ ge \ ge \ ge \ frac {d} {d} {2} {2} {2} $。 First, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian $(-Δ)^{s/2}$ with $s\in (0,1]$ established by [Discrete Contin. Dyn. Syst. 41 (2021) 5409-5437] to any $s>0$. Applying this estimate and the contraction mapping principle基于洛伦兹空间中的估计，我们在$ h^{s} $中以$ h^{s} $建立了IBNLS方程，在两个亚临界情况下，$σ<σ_{c}（c}（s）$和关键情况$σ=σ_{c}（c）初始数据足够小，$ \ frac {8-2b} {d} \ leσ\leσ_{c} {c}（s）$，带有$σ<\ infty $。
We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,\;u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb R$, $d\in \mathbb N$, $0\le s<\min\left\{2+\frac{d}{2},d\right\}$, $0<b<\min \left\{4,\; d-s,\; 2+\frac{d}{2}-s \right\}$ and $0<σ\le σ_{c}(s)$ with $σ<\infty$. Here $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. First, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian $(-Δ)^{s/2}$ with $s\in (0,1]$ established by [Discrete Contin. Dyn. Syst. 41 (2021) 5409-5437] to any $s>0$. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in $H^{s}$ for the IBNLS equation in both of subcritical case $σ<σ_{c}(s)$ and critical case $σ=σ_{c}(s)$. We also prove that the IBNLS equation is globally well-posed in $H^{s}$, if the initial data is sufficiently small and $\frac{8-2b}{d}\le σ\le σ_{c}(s)$ with $σ<\infty$.
