Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dissipation and Regularity in the Smagorinsky Model with Dynamic Boundaries

Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales

Abstract

This article presents an innovative extension of the Smagorinsky model incorporating dynamic boundary conditions and advanced regularity methods. We formulate the modified Navier-Stokes equations with the Smagorinsky term to model dissipation in turbulence and prove theorems concerning the existence, uniqueness, and asymptotic behavior of solutions. The first theorem establishes the existence and uniqueness of solutions in higher Sobolev spaces, considering the effect of the nonlinear Smagorinsky term and dynamic boundary conditions. The proof employs the Galerkin method and energy estimates, culminating in the application of Grönwall's theorem. The second theorem investigates the asymptotic behavior of solutions, focusing on anomalous dissipation in high-turbulence regimes. We demonstrate that the dissipated energy does not decrease with vanishing viscosity, indicating the occurrence of anomalous dissipation. The third theorem explores advanced regularity in higher Sobolev spaces, allowing for more rigorous control of nonlinear terms and ensuring improved stability conditions. The proof utilizes the energy method combined with Sobolev estimates and Grönwall's inequality. These mathematical results are fundamental for the analysis of dissipation in turbulent flows and can inspire new approaches in numerical simulations of fluids.

Dissipation and Regularity in the Smagorinsky Model with Dynamic Boundaries

Abstract

This article presents an innovative extension of the Smagorinsky model incorporating dynamic boundary conditions and advanced regularity methods. We formulate the modified Navier-Stokes equations with the Smagorinsky term to model dissipation in turbulence and prove theorems concerning the existence, uniqueness, and asymptotic behavior of solutions. The first theorem establishes the existence and uniqueness of solutions in higher Sobolev spaces, considering the effect of the nonlinear Smagorinsky term and dynamic boundary conditions. The proof employs the Galerkin method and energy estimates, culminating in the application of Grönwall's theorem. The second theorem investigates the asymptotic behavior of solutions, focusing on anomalous dissipation in high-turbulence regimes. We demonstrate that the dissipated energy does not decrease with vanishing viscosity, indicating the occurrence of anomalous dissipation. The third theorem explores advanced regularity in higher Sobolev spaces, allowing for more rigorous control of nonlinear terms and ensuring improved stability conditions. The proof utilizes the energy method combined with Sobolev estimates and Grönwall's inequality. These mathematical results are fundamental for the analysis of dissipation in turbulent flows and can inspire new approaches in numerical simulations of fluids.

Paper Structure

This paper contains 16 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Background in Mathematics
  3. Sobolev Spaces
  4. Energy Methods
  5. Grönwall's Inequality
  6. Physical and Mathematical Formulation
  7. Theorem 1: Existence and Uniqueness of Solutions
  8. Theorem 2: Asymptotic Behavior and Anomalous Dissipation
  9. Theorem 3: Advanced Regularity in Sobolev Spaces
  10. Results
  11. Existence and Uniqueness of Solutions
  12. Asymptotic Behavior and Anomalous Dissipation
  13. Advanced Regularity in Sobolev Spaces
  14. Nomenclature and Symbols
  15. Nomenclature
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Omega$ be a bounded domain with a smooth boundary and $f \in L^2(0, T; H^{-1}(\Omega))$. There exists a unique solution $u \in L^2(0, T; H^s(\Omega)) \cap C([0, T]; L^2(\Omega))$ for equation eq:smagorinsky_model, satisfying: where $C$ depends on the initial data and $f$.

Theorems & Definitions (6)

  • Theorem 1: Existence and Uniqueness in $H^s(\Omega)$
  • proof
  • Theorem 2: Asymptotic Behavior and Anomalous Dissipation
  • proof
  • Theorem 3: Advanced Regularity
  • proof