arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1601.06019math.AP
收藏

$L^{p}$ -空间上具有Navier型边界条件的Stokes算子的群论

Semi-group theory for the Stokes operator with Navier-type boundary conditions on $L^{p}$-spaces

Hind Al Baba, Chérif Amrouche, Miguel Escobedo

摘要:

在这篇文章中,我们考虑的Stokes问题的Navier型边界条件的区域$Ω$,不一定单连通。由于在这些条件下的Stokes问题有一个非平凡的内核,我们还研究了该内核的正交的解决方案。证明了由Stokes算子生成的几个半群在不同的泛函空间中的解析性。在对初值$\boldsymbol{u}_0$和外力$\boldsymbol{f}$的不同假设下,得到了具有Navier型边界条件的含不定常Stokes问题的强、弱和很弱解.然后,我们研究了与我们的Stokes算子相关的几个算子的分数幂和纯虚幂。利用分数幂,我们证明了齐次Stokes问题的极大正则性结果。另一方面,利用纯虚幂的有界性,我们推导出非齐次Stokes问题的极大$L^{p}-L^{q}$正则性。

英文摘要:

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $Ω$, not necessarily simply connected. Since under these conditions the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypothesis on the initial data $\boldsymbol{u}_0$ and external force $\boldsymbol{f}$. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers we deduce maximal $L^{p}-L^{q}$ regularity for the inhomogeneous Stokes problem.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

↑