论文标题
与球形对称性的准线性波方程的尖锐局部适合度
Sharp local well-posedness for quasilinear wave equations with spherical symmetry
论文作者
论文摘要
在本文中，当空间尺寸为三个或更高时，我们证明了针对具有粗糙初始数据的准电波方程的球形对称解的局部局部良好结果。我们的方法是基于莫拉维兹类型的局部能量估计值，具有分数规则性的线性波方程，具有可变的$ c^1 $系数，这些系数依赖于乘数方法，加权的利特伍德 - 帕利理论，二元性和插值。适用于问题的加权线性和非线性估计（包括加权痕量估计，Hardy的不平等，分数链规则和分数Leibniz规则），可以通过迭代证明了良好的结果。此外，我们的论点几乎以$ n = 3 $而产生的全球存在，而全球存在则为$ n \ ge 4 $，当初始数据很小时，球形对称，几乎具有至关重要的Sobolev规律性。
In this paper, we prove a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type local energy estimates with fractional regularity for linear wave equations with variable $C^1$ coefficients, which rely on multiplier method, weighted Littlewood-Paley theory, duality and interpolation. Together with weighted linear and nonlinear estimates (including weighted trace estimates, Hardy's inequality, fractional chain rule and fractional Leibniz rule) which are adapted for the problem, the well-posed result is proved by iteration. In addition, our argument yields almost global existence for $n=3$ and global existence for $n\ge 4$, when the initial data are small, spherically symmetric with almost critical Sobolev regularity.
