Global in Time Solution to Kolmogorov’s Two-equation Model of Turbulence with Small Initial Data

Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces

“…In particular, in [17] (see also [16] for an announcement of the result), Mielke and Naumann proved the existence of global in time finite energy weak solutions to (1) in the periodic three dimensional box T 3 , under condition (2). The same condition was used by Kosewski and Kubica to set down a strong solutions theory, see [11] and [12] for, respectively, a local well-posedness result and a global well-posedness result for small initial data (see also [10] by Kosewski for an extension to the case of fractional regularities).…”

Finite time blow-up for some parabolic systems arising in turbulence theory

“…If the latter issue is a key property any model of turbulence should retain, the former represents instead a delicate point for the mathematical analysis. As a matter of fact, if we restrict our attention to the Kolmogorov model (3) for instance, previous results on well-posedness in the framework of both weak and strong solutions have always either avoided to consider the possible vanishing of the function k (see [21], [11] and [12], for instance), or imposed suitable conditions on the initial data to control the way k may get close to the 0 value (see for instance [4] in this direction).…”