Local well-posedness of Kolmogorov's two-equation model of turbulence in fractional Sobolev Spaces

TL;DR

This paper proves the local existence and uniqueness of solutions to Kolmogorov's two-equation turbulence model in fractional Sobolev spaces, expanding understanding of well-posedness in mathematical turbulence theory.

Contribution

It establishes local well-posedness of the model in fractional Sobolev spaces for the first time, using energy methods.

Findings

Solutions exist locally in time for initial data in H^s with s > d/2.

Solutions are unique within the class of solutions constructed.

The analysis is conducted on the d-dimensional torus.

Abstract

We study Kolmogorov's two-equation model of turbulence on $d-$dimensional torus. First, the local existence of the solution with the initial data from non-homogeneous fractional Sobolev spaces (Bessel potential spaces) $H^s$ with $s>\frac{d}{2}$ is proven using energy methods. Next, we show that solutions are unique in the class of solutions guaranteed by the local existence theorem.