验证减少阶替代方案与粘性不可压缩流体及流体-结构相互作用中的稳定性边界
Certified Reduced-Order Surrogates and Stability Margins in Viscous Incompressible Flow and Fluid--Structure Interaction
AI总结：
本文提出了一种可验证的减少阶模型，用于粘性不可压缩流体及流体-结构相互作用中的稳定性分析，通过能量不等式和残差泛函提供稳定性边界。
摘要：
设$(u,p)$满足不可压缩纳维-斯托克斯方程，且在某一能量不等式成立的区域内，每个不等式中的常数均可从声明的数据中计算得出。我们构造了一个减少阶模型$u_n$，使其离散演化满足一个可验证的能量不等式。该证书可以提供全局时间内的减少阶模型能量有界性，并提供一个有效性区域测试，当所假设的假设不成立时，该测试会失败。由此可以附加一个可计算的残差泛函$\mathcal{R}_n$到减少阶模型轨迹上。我们证明了一个后验界形式为$$\norm{u-u_n}_{\mathsf{X}(0,T)} \le C(\text{declared data})\,\mathcal{R}_n，$$其中$C$是显式且$\mathcal{R}_n$由减少阶模型和离散化算子计算得出。反之，如果放松证书约束，即使对于稳定的全阶动态，该界也可能失效，通过文本中记录的显式不稳定性机制。然后我们从简化几何中导出转换指示器，每个指示器都是涉及声明量的不等式，如力的范数、粘度、皮尔卡尔型常数以及可计算的共振替代方案。这些不等式提供了防止转换的阈值，或者认证在所陈述水平以上的瞬态增长的存在。最后，对于一类流体-结构相互作用模型，我们识别出一个参数区域，该区域暗示了弱解的存在性和唯一性。我们导出了离散耦合能量估计，这些估计产生可计算的稳定性边界。这些边界提供了对时间步长和网格参数的显式约束。它们以包含由流体粘度、结构刚度、密度比和界面迹界确定的常数的不等式形式给出。
英文摘要：
Let $(u,p)$ solve the incompressible Navier--Stokes equations in a regime in which an energy inequality is available and each constant in that inequality is computable from declared data. We construct a reduced-order model $u_n$ constrained so that its discrete evolution satisfies a certified energy inequality. This certificate yields global-in-time boundedness of the ROM energy and a regime-of-validity test that fails when a stated hypothesis fails. It follows that one can attach a computable residual functional $\mathcal{R}_n$ to the ROM trajectory. We prove an a posteriori bound of the form \[ \norm{u-u_n}_{\mathsf{X}(0,T)} \le C(\text{declared data})\,\mathcal{R}_n, \] with $C$ explicit and with $\mathcal{R}_n$ computed from the ROM and the discretization operators. Conversely, if the certificate constraint is relaxed, the bound can fail even for stable full-order dynamics, by an explicit instability mechanism recorded in the text. We then derive transition indicators from rigorous energy and enstrophy budgets in simplified geometries. Each indicator is an inequality involving declared quantities such as forcing norms, viscosity, Poincaré-type constants, and a computable resolvent surrogate. These inequalities provide thresholds that preclude transition, or else certify the presence of transient growth beyond a stated level. Finally, for a class of fluid--structure interaction models, we identify a parameter regime that implies existence and uniqueness of weak solutions. We derive discrete coupled energy estimates that produce computable stability margins. These margins yield explicit constraints on time step and mesh parameters. They are stated as inequalities with constants determined by fluid viscosity, structure stiffness, density ratios, and interface trace bounds.
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。
