Dissipation and Regularity in the Smagorinsky Model with Dynamic Boundaries

TL;DR

This paper extends the Smagorinsky turbulence model with dynamic boundary conditions, proving new theorems on solution existence, uniqueness, asymptotic behavior, and regularity, which enhance understanding of dissipation in turbulent flows.

Contribution

It introduces a modified Navier-Stokes framework with the Smagorinsky term and dynamic boundaries, providing rigorous mathematical analysis of solutions and dissipation phenomena.

Findings

Proved existence and uniqueness of solutions in higher Sobolev spaces.

Demonstrated anomalous dissipation in high-turbulence regimes.

Established improved regularity and stability conditions.

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 Gr\"onwall'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 Gr\"onwall'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.