论文标题
在无限质量和低规律性的空间中的非线性schrödinger方程上
On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity
论文作者
论文摘要
我们在$ \ Mathcal {z}^s_p（\ Mathbb {r}^d）= \ dot {h}^s（\ Mathbb {r}^d）\ cap l^p（\ mathbb {r}^d d）$ 0 <s <s <s <s <s <s（ $ 2 <p <2d/（d-2s）$。在证明线性schrödinger组在这个领域的定义明确后，我们证明了整个参数范围内的本地适合度，$ s $ s $和$ p $。解决方案的精确属性取决于非线性功率与集成性$ p $之间的关系。最后，我们使用傅立叶截断方法的变体，在维焦点方程中散发出一个全球存在的结果。
We study the nonlinear Schrödinger equation with initial data in $\mathcal{Z}^s_p(\mathbb{R}^d)=\dot{H}^s(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$, where $0<s<\min\{d/2,1\}$ and $2<p<2d/(d-2s)$. After showing that the linear Schrödinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.
