Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Manuel Fernando Cortez, Oscar Jarrin

TL;DR

This work provides a rigorous mathematical analysis of a novel Navier–Stokes–alpha type model featuring a nonlinear filter that adapts eddy damping via an indicator $A$. The authors establish global existence of weak Leray solutions, derive a conditional uniqueness result under regularity assumptions on $A$ and the mollifier $\\varphi$, and prove convergence to classical models as $\\alpha\\to0$ (NS) and $\\beta\\to1$ (Leray–alpha). They also study the long-time behavior by constructing a global attractor and giving explicit, parameter-dependent upper bounds on its fractal dimension, highlighting how forcing and regularization parameters influence attractor size. The results rely on a regularized filter equation to secure $H^2$ control and use periodic boundary conditions to exploit Fourier-analytic techniques, yielding a coherent theoretical justification for the alpha-model as a robust regularization of NS with meaningful turbulence-relevant dynamics.

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.

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

TL;DR

This work provides a rigorous mathematical analysis of a novel Navier–Stokes–alpha type model featuring a nonlinear filter that adapts eddy damping via an indicator . The authors establish global existence of weak Leray solutions, derive a conditional uniqueness result under regularity assumptions on and the mollifier , and prove convergence to classical models as (NS) and (Leray–alpha). They also study the long-time behavior by constructing a global attractor and giving explicit, parameter-dependent upper bounds on its fractal dimension, highlighting how forcing and regularization parameters influence attractor size. The results rely on a regularized filter equation to secure control and use periodic boundary conditions to exploit Fourier-analytic techniques, yielding a coherent theoretical justification for the alpha-model as a robust regularization of NS with meaningful turbulence-relevant dynamics.

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.
Paper Structure (16 sections, 27 theorems, 291 equations)

This paper contains 16 sections, 27 theorems, 291 equations.

Table of Contents

  1. Introduction and motivation of the model
  2. Main results
  3. An example of a convolution term
  4. The regularized nonlinear filter equation with $L^2$-data
  5. Global well-posedness of weak Leray solutions
  6. Proof of Theorem \ref{['Th-Existence']}
  7. Proof of Theorem \ref{['Th-Uniqueness']}
  8. Convergence properties to classical models
  9. Proof of Proposition \ref{['Prop1-Conv-NS']}
  10. Proof of Proposition \ref{['Prop2-Conv-NSalpha']}
  11. Global attractor and fractal dimension
  12. Existence of a global attractor: Proof of Theorem \ref{['Th-Global-Attractor']}
  13. Upper bounds of the fractal dimension: proof of Theorem \ref{['Th-Fractal-Dimension']}
  14. The non-regularized nonlinear filter equation
  15. Acknowledgements.
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\vec{u}_0 \in L^2_{\text{s}}(\mathbb{T}^3)$ be an initial datum, and let $\vec{f} \in L^{2}_{loc}([0,+\infty[, \dot{H}^{-1}(\mathbb{T}^3))$ be a divergence-free and zero-mean external force. In the nonlinear filter equation of the system (Alpha-model) assume that the indicator function $A(\cdot Moreover, assume that the convolution term $\varphi$ verifies: Then, for every $\alpha>0$ the coup

Theorems & Definitions (38)

  • Definition 2.1
  • Remark 1
  • Theorem 2.1: Existence of weak Leray solutions
  • Theorem 2.2: Uniqueness
  • Proposition 2.1: Convergence to the Navier-Stokes equations
  • Proposition 2.2: Convergence to the Leray-alpha model
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.3: Existence of a unique global attractor
  • Definition 2.4
  • ...and 28 more