Mathematical study of a new Navier-Stokes-alpha model with nonlinear filter equation -- Part I

TL;DR

This paper provides a rigorous mathematical analysis of a new Navier-Stokes-alpha model with a nonlinear filter, establishing existence, uniqueness, convergence, and long-term behavior of solutions under periodic boundary conditions.

Contribution

It is the first to analyze the doubly nonlinear coupled system of this new alpha-model, including solutions' existence, uniqueness, convergence, and attractor properties.

Findings

Proved existence and uniqueness of weak solutions.

Established convergence to classical Navier-Stokes solutions.

Analyzed long-time dynamics and estimated fractal dimension of attractor.

Abstract

This article is devoted to the mathematical study of a new Navier-Stokes-alpha model with a nonlinear filter equation. For a given indicator function, this filter equation was first considered by W. Layton, G. Rebholz, and C. Trenchea to select eddies for damping based on the understanding of how nonlinearity acts in real flow problems. Numerically, this nonlinear filter equation was applied to the nonlinear term in the Navier-Stokes equations to provide a precise analysis of numerical diffusion and error estimates. Mathematically, the resulting alpha-model is described by a doubly nonlinear parabolic-elliptic coupled system. We therefore undertake the first theoretical study of this system by considering periodic boundary conditions in the spatial variable. Specifically, we address the existence and uniqueness of weak Leray-type solutions, their rigorous convergence to weak Leray solutions of the classical Navier-Stokes equations, and their long-time dynamics through the concept of the global attractor and some upper bounds for its fractal dimension. Handling the nonlinear filter equation together with the well-known nonlinear transport term makes certain estimates delicate, particularly when deriving upper bounds on the fractal dimension. For the latter, we adapt techniques developed for hyperbolic-type equations.