论文标题
在Besov-Morrey空间中的本地和全球分析，用于不均匀的Navier-Stokes方程
Local and global analysis in Besov-Morrey spaces for inhomogeneous Navier-Stokes equations
论文作者
论文摘要
在本文中，我们考虑了整个空间中不可压缩的不均匀的Navier-Stokes方程，并具有$ n \ geq 3 $。我们介绍了本地和全球良好的性能，为不均匀的流体带来了一个新的框架，即besov-morrey空间$ \ mathcal {n} _ {p，q，q，q，r}^{s} $是基于morrey的besov space。与先前在Sobolev和Besov空间中的作品相比，我们的结果为速度和密度提供了更大的初始数据类别，并在较弱的初始数据规范的情况下构建了独特的全球时间流。特别是，我们可以考虑某种初始不连续的密度，因为我们的密度类别$ \ MATHCAL {n} _ {p，q，q，\ infty}^{n/p} \ cap l^{\ infty} $都不包含在连续功能的任何空间中。从技术角度来看，莫雷的基本规范可以阻止能量类型的常见使用和零件参数的整合，然后我们需要获得一些估计热半群，换向器的本地化，换向器以及在我们的设置中的音量扩散图，以及对运输方程和线性的Inhomogeenized Inhomogeensemenoseverenos navierous-Stokes Systems的估算。
In this paper we consider the incompressible inhomogeneous Navier-Stokes equations in the whole space with dimension $n\geq 3$. We present local and global well-posedness results in a new framework for inhomogeneous fluids, namely Besov-Morrey spaces $\mathcal{N}_{p,q,r}^{s}$ that are Besov spaces based on Morrey ones. In comparison with the previous works in Sobolev and Besov spaces, our results provide a larger initial-data class for both the velocity and density, constructing a unique global-in-time flow under smallness conditions on weaker initial-data norms. In particular, we can consider some kind of initial discontinuous densities, since our density class $\mathcal{N}_{p,q,\infty }^{n/p}\cap L^{\infty }$ is not contained in any space of continuous functions. From a technical viewpoint, the Morrey underlying norms prevent the common use of energy-type and integration by parts arguments, and then we need to obtain some estimates for the localizations of the heat semigroup, the commutator, and the volume-preserving map in our setting, as well as estimates for transport equations and the linearized inhomogeneous Navier-Stokes system.
