Certified Reduced-Order Surrogates and Stability Margins in Viscous Incompressible Flow and Fluid--Structure Interaction

Abstract

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.