定常纳维-斯托克斯流固耦合问题的数据到解映射的解析性
Analyticity of the data-to-solution map for a stationary Navier-Stokes fluid-structure interaction problem
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中文总结 AI 辅助
该研究针对定常纳维-斯托克斯与欧拉-伯努利梁耦合的流固问题，通过复化方法和全纯隐函数定理证明数据到解映射的实解析性，给出小数据存在性与局部唯一性结果，相关结论可用于参数偏微分方程的数据驱动降阶建模。
中文摘要 AI 辅助
我们研究一个定常流固耦合问题，该问题中定常纳维-斯托克斯方程通过一个自由弹性界面与夹紧的欧拉-伯努利梁方程耦合。利用固定域形式的复化方法和全纯隐函数定理，我们证明在平凡解的邻域内，从右端项到弱解的映射是实解析的。作为推论，我们得到了该耦合系统的小数据存在性和局部唯一性结果。我们的研究动机来自参数偏微分方程的数据驱动降阶建模，其中近似特性与解映射的正则性密切相关。数值上，一个 manufactured-solution 测试显示其报告的相对L²误差近似二阶收敛，而对参数力族的本征正交分解研究显示经验重构误差快速衰减直至达到数值下限。
英文摘要
We consider a stationary fluid--structure interaction problem in which the steady Navier--Stokes equations are coupled, through a free elastic interface, with a clamped Euler-Bernoulli beam equation. Using a complexification of the fixed-domain formulation and the holomorphic implicit function theorem, we prove that, in a neighbourhood of the trivial solution, the mapping from the right-hand side to the weak solution is real analytic. As a byproduct we obtain a small-data existence and local uniqueness result for the coupled system. Our motivation comes from data-driven reduced-order modelling for parametric PDEs, where approximation properties are closely related to the regularity of the solution map. Numerically, a manufactured-solution test exhibits approximately second-order convergence in the reported relative $L^2$ errors, while a proper orthogonal decomposition study for a parametric force family shows rapid decay of the empirical reconstruction error until a numerical floor is reached.
