齐次加权函数空间中外区域上的Stokes算子：从弱理论到$\u2708^\u221e$-演算再到分数次定义域

中文摘要 AI 辅助

我们研究$\u211d^n$中光滑外区域$\u03a9$上、带径向对称Muckenhoupt权$w\in \mathscr A_q$的齐次Sobolev空间$\u02c6 H^{\u03ba,q}_w(\u03a9)$中的Stokes算子$A$。核心性质是Stokes算子在加权非齐次与齐次$L^q$ Sobolev空间上存在有界$\u2708^\u221e$-演算。该性质意味着一致有界纯虚幂$A^{it}$（$t\in\mathbb{R}$）的存在性，且装备非齐次范数（$\\| u\\|_{L^q_w} + \\|A^\u03b8u\\|_{L^q_w}$）与齐次范数（$\\|A^\u03b8u\\|_{L^q_w}$）的分数次幂$A^\u03b8$的定义域可刻画为复插值空间。最终目标是将其与无散向量场空间相交的齐次空间$\u02c6{\mathcal D}((-\u0394_{q,w})^\u03b8) = [L^q_{w},\u02c6{\mathcal D}(-\u0394_{q,w})]_\u03b8$建立等同关系。此外，我们还得到了Stokes方程弱解的加权变分不等式、Stokes半群的加权$L^q$-$L^r$衰减估计，以及$L^q_{\u03c3,w}(\u03a9)$上的$L^p$-极大正则性。

英文摘要

We consider the Stokes operator $A$ on smooth exterior domains $Ω$ of $\mathbb{R}^n$ in homogeneous Sobolev spaces $\widehat H^{κ,q}_w(Ω)$ with radially symmetric Muckenhoupt weights $w\in \mathscr A_q$. A fundamental property is the existence of a bounded $\mathscr H^\infty$-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous $L^q$ Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers $A^{it}$, $t\in\mathbb{R}$, and the characterization of domains of fractional powers $A^θ$ equipped with nonhomogeneous ($\| u\|_{L^q_w} + \|A^θu\|_{L^q_w}$) as well as homogeneous norm ($\|A^θu\|_{L^q_w}$) as complex interpolation spaces. The final aim is the identification with homogeneous spaces $\widehat{\mathcal D}((-Δ_{q,w})^θ) = [L^q_{w},\widehat{\mathcal D}(-Δ_{q,w})]_θ$ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted $L^q$-$L^r$ decay estimates of the Stokes semigroup and $L^p$-maximal regularity on $L^q_{σ,w}(Ω)$.

发表机构

• Technische Universität Darmstadt（达姆施塔特工业大学）
• Kyushu Sangyo University（九州产业大学）

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