论文标题
关于具有部分规则数据的非线性色散方程的本地良好性
On local well-posedness of nonlinear dispersive equations with partially regular data
论文作者
论文摘要
我们重新访问sobolev spaces $ h^s $和$ \ dot {h}^s $，$ 0 <s \ leq 1 $中的非线性schrödinger和波动方程的本地稳定性理论。在过去的几十年中，在Sobolev初始数据规则的所有空间变量下，该理论已经建立了很好的确定。但是在这里，我们揭示了初始数据不需要像Sobolev空间那样具有完整的规律性，但是对于某些变量，只有部分规律性就足够了。为了发展这样的新理论，我们建议一个精致的strichartz估计值，该估计具有不同的规范。这使得可以从每个变量中提取数据的不同集成性/规律性。
We revisit the local well-posedness theory of nonlinear Schrödinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data regular with respect to all spatial variables. But here, we reveal that the initial data do not need to have complete regularity like Sobolev spaces, but only partially regularity with respect to some variables is sufficient. To develop such a new theory, we suggest a refined Strichartz estimate which has a different norm for each spatial variable. This makes it possible to extract a different integrability/regularity of the data from each variable.
