Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence

Abstract

This paper is concerned with Kolmogorov's two-equation model for free turbulence in R 3 involving the mean velocity u, the pressure p, an average frequency ω > 0, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under spaceperiodic boundary conditions in cubes Ω = ( ] 0, l [ ) 3 with l > 0. For this we provide new a priori estimates and invoke existence result for pseudo-monotone operators. * A.M. was supported by Deutsche Forschungsgemeinschaft (DFG) via project A5 within SFB 910.

Thermoviscoelasticity in Kelvin–Voigt Rheology at Large Strains

“…Proof. We follow the recipe in [11] in the simplified variant of [22], see also [33]. For η ∈ ]0, 1[ we define the function χ η : 1b) with the test function z = χ η • w ε amounts to applying the chain rule in Proposition 3.5 to the convex functional J (w) = Ω χ η (w(x)) dx on the space X = H 1 (Ω) * .…”

“…where crucially relied on p = 2/r , χ (w) = η/(1+w) 1+η , and the previous estimate. Using the a priori estimate 1+w ε L ∞ (I ;L 1 (Ω)) T |Ω| + K =: K 1 from (6.2c) we can now use the anisotropic Gagliardo-Nirenberg interpolation (see for example [33,Lem. 4…”

Local in time solution to Kolmogorov's two-equation model of turbulence

“…Now, we would like to discus the known mathematical results related to Kolmogorov's two-equation model of turbulence. There are two recent results devoted to this problem: [1] and [3] (see also the announcement [4]) and our result is inspired by them. In the first one, the Authors consider the system in a bounded C 1,1 domain with mixed boundary conditions for b and ω and stick-slip boundary condition for velocity v. In order to overcome the difficulties related with the last term on the right hand side of (3) the problem is reformulated and the quantity E := 1 2 |v| 2 + 2ν 0 κ 4 b is introduced.…”

Well-posedness and singularity formation for the Kolmogorov two-equation model of turbulence in 1-D

“…In [13] (see also [12] for an announcement of the results), Mielke and Naumann proved the existence of global in time weak solutions to system (1) set on the torus T 3 . The solutions constructed therein are analogous in spirit to Leray's finite energy solutions for the incompressible Navier-Stokes equations, except for the fact that k cannot really belong to L 2 , because of the last term appearing on the right-hand side of its equation, which is merely in L 1 (yet, the authors are able to prove a gain of integrability for that quantity).…”