On the existence of weak solutions for a family of unsteady rotational Smagorinsky models

TL;DR

This paper establishes the existence of weak solutions for a class of unsteady rotational Smagorinsky models in turbulent flow, using Bochner pseudo-monotone evolution equations and analyzing the role of model exponents.

Contribution

It introduces a novel functional framework for the rotational Smagorinsky model and proves weak solution existence for a broad parameter range.

Findings

Existence of weak solutions in weighted spaces.

Identification of key parameters affecting solution existence.

Analysis of the impact of model exponents on solution theory.

Abstract

In this paper we show that the rotational Smagorinsky model for turbulent flows, can be put, for a wide range of parameters in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions a) identifying a proper functional setting in weighted spaces and b) checking some easily verifiable assumptions, at fixed time. We also will discuss the critical role of the exponents present in the model (power of the distance function and power of the curl) for what concerns the application of the theory of pseudo-monotone operators.