论文标题
Stokes操作员在二维有限的Lipschitz域中
The Stokes operator in two-dimensional bounded Lipschitz domains
论文作者
论文摘要
我们考虑在二维有限的Lipschitz域$ω$的stokes分解问题。我们证明$ \ mathrm {l}^p $ -resolvent估计$ p $满足条件$ \ lvert 1 / p -1 / p -1/2 \ rvert <1/4 + \ varepsilon $，用于某些$ \ varepsilon> 0 $。我们进一步表明，stokes操作员承认了最大规律性的属性，并且其$ \ mathrm {h}^{\ infty} $ - 计算是有限的。然后将其用于表征Stokes操作员分数幂的域。最后，我们将弱解决方案的规律性理论应用于有限的平面Lipschitz域中的Navier-Stokes方程。
We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $Ω$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$. We further show that the Stokes operator admits the property of maximal regularity and that its $\mathrm{H}^{\infty}$-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutions to the Navier-Stokes equations in bounded planar Lipschitz domains.
