Gevrey class regularity for steady-state incompressible Navier-Stokes equations in parametric domains and related models

TL;DR

This work develops a Gevrey-$G^{ abla}$ regularity theory for parametric, steady incompressible Navier–Stokes equations on domains perturbed by high-dimensional random inputs under a small-data regime. By employing an alternative-to-factorial approach and a detailed treatment of mixed/saddle-point formulations, it derives explicit high-order derivative bounds for velocity and pressure with respect to the parametric variables, and shows that the analytic case ($ abla=1$) is recovered as a special instance. The authors extend Gevrey regularity through domain transform pullbacks and provide sharp estimates for derivatives of matrix-valued mappings, using Faà di Bruno-type decompositions to control composite dependencies. Numerical experiments using Gauss-Legendre quadrature and randomly shifted lattice QMC corroborate the theoretical rates, highlighting enhanced convergence for analytic perturbations and quantified decay for Gevrey perturbations. The results have significant implications for uncertainty quantification in fluid dynamics, enabling rigorous, high-accuracy evaluation of parametric statistics via fast quadrature schemes.

Abstract

We investigate parameteric Navier-Stokes equations for a viscous, incompressible flow in bounded domains. The coefficients of the equations are perturbed by high-dimensional random parameters, this fits in particular for modelling flows in domains with uncertain perturbations. Our focus is on deriving bounds for arbitrary high-order derivatives of the pressure and the velocity fields with respect to the random parameters in the context of incompressible Navier-Stokes equation under a small-data assumption. To achieve this, we analyze mixed and saddle-point problems and employ the alternative-to-factorial technique to establish generalized Gevrey-class regularity for the solution pair. Thereby the analytic regularity follows as a special case. In the numerical experiments, we validate and illustrate our theoretical findings using Gauss-Legendre quadrature and Quasi-Monte Carlo methods.

Figures (4)

• Figure 1: The absolute error of Gauss-Legendre quadrature: Navier-Stokes equation
• Figure 2: Pressure and streamlines of velocity field solution of Navier-Stokes equation for $T^{(1)}$ at $y=1$
• Figure 3: Pressure and streamlines of velocity field solution of Navier-Stokes equation for $T^{(2)}$ at $y=0.76$
• Figure 4: Convergence of the quadrature error with respect to the number of samples $n$ for four methods: QMC analytic ($\varepsilon^{{\rm QMC},(1)}_n$), QMC Gevrey ($\varepsilon^{{\rm QMC},(2)}_n$), MC analytic ($\varepsilon^{{\rm MC},(1)}_n$), MC Gevrey ($\varepsilon^{{\rm MC},(2)}_n$).

Theorems & Definitions (53)

• Example 1: Parametric Domain Perturbations
• Example 2: Stochastic Steady-State Navier-Stokes Flow
• Lemma 1
• Definition 1
• Remark 1
• Remark 2
• Proposition 1
• proof
• Remark 3
• Lemma 2