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

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

“…The same positivity assumption (3) is formulated (and plays a fundamental role) in works [8] and [9] by Kosewski and Kubica. There, the authors prove respectively local in time wellposedness, global in time well-posedness for small data, for the Kolmogorov model (1) set in T 3 , at H 2 level of regularity on the initial data.…”

“…We approximate problem (4) with vanishing initial k 0 with problems having initial k 0,ε 's which satisfy k 0,ε ≥ ε. We thus construct approximate solutions by using the result from [8], and we pass to the limit in the approximation parameter ε → 0 + by using a compactness argument. This proves the existence, for which we refer to Subsection 4.1.…”

“…By adapting, for instance, the method of [8] to the one-dimensional setting, we construct a local in time solution u ε , ω ε , k ε at the H 2 level of regularity, defined in some time interval [0, T ε ] (see Theorem 1 in [8] for a precise statement).…”

“…The construction of solutions in [8] passes through Galerkin approximation. Thus, the approximated solutions are smooth, and our computations of Section 3 are fully justified for them.…”

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 [13], [9] and [10], 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 [2] in this direction).…”