AI 中文总结
研究外区域中纳维 - 斯托克斯方程全局弱解的\(L^p\)理论，通过特定方法证明其存在性并给出结构定理，表明弱解在一定时间后正则，此理论在相关特殊问题中也适用，为该领域提供了新成果。
AI 中文摘要
本文考虑外区域中具有\(L^p\)空间（\(p\in(2,3)\)）初始数据的纳维 - 斯托克斯初边值问题，证明了全局弱解的存在性。对于该解，还给出了一个结构定理。特别地，弱解在某一时刻后变得正则，且在几乎所有时间上正则。虽然局部强/温和解的一般\(L^p\)理论已成熟，但外区域全局弱解的相应理论似乎是新的。结果在柯西问题和半空间初边值问题的特殊情况下也成立。
英文摘要
In this paper, we consider the Navier-Stokes initial boundary value problem in exterior domains with initial data in the Lebesgue space $L^p$, $p\in (2,3)$, and show the existence of a global weak solution. For the quoted solution, we also furnish a structure theorem. In particular, we see that the weak solution becomes regular after a certain instant of time and it is also regular a. e. in time. Although a general $L^p$-theory for local strong/mild solutions is well-established in the literature, a corresponding theory for global weak solutions in exterior domains seems to be new. Of course, our results hold in the particular cases of the Cauchy problem and the initial boundary value problem in a half-space.
