具周期运动边界的外域中Navier-Stokes方程的时间周期问题
Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary
- 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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AI总结：
本文研究带时间周期运动边界的外域Navier-Stokes方程，在加权函数空间中证明了局部唯一温和时间周期解的存在性，采用了解析半群理论、Stokes算子的ℋ^∞演算等方法，相关成果发表于2025年的论文中。
AI中文摘要：
本文研究n≥3维欧氏空间ℝⁿ外域中带时间周期运动边界∂Ω(t)和外力f(t)的Navier-Stokes方程，在带径向对称Muckenhoupt权的加权函数空间中证明了局部唯一的温和时间周期解的存在性。该解分为两部分：由位势论估计控制的定常部分，以及通过解析半群理论构造为温和解的纯振荡部分。为处理坐标变换和运动边界带来的加权齐次Sobolev空间中偶二阶摄动项，需在加权Lorentz空间中采用极大L¹型正则性估计；为控制对流项，利用Stokes算子在加权空间中的ℋ^∞演算、其BIP性质及分数幂的嵌入估计，相关结果参见作者2025年的最新论文《外域齐次加权函数空间中的Stokes算子：从弱理论到ℋ^∞演算再到分数阶域》。
英文摘要：
In this paper we consider the Navier-Stokes equations in exterior domains of $\mathbb{R}^n$, $n\geq 3$, with a periodically in time moving boundary $\partialΩ(t)$ and external force $f(t)$. For this case we prove the existence of a locally unique mild time periodic solution in weighted function spaces with radially symmetric Muckenhoupt weights. The solutions split into a stationary part controlled by potential theoretic estimates and a purely oscillatory part constructed as mild solution via analytic semigroup theory. To deal with perturbation terms of even second order - coming from a coordinate transform and the moving boundary - in weighted, homogeneous Sobolev spaces a maximal $L^1$ type regularity estimate will be used in weighted Lorentz spaces. To control the convective term an $\mathcal H^\infty$-calculus in weighted spaces of the Stokes operator, its $BIP$ property and embedding estimates of fractional powers are exploited, see a recent paper by the authors: The Stokes operator on exterior domains in homogeneous weighted function spaces: From weak theory to $\mathscr H^\infty$-calculus to fractional domains (2025).
