Parameter dependence in the \(b-\varepsilon\) model (Q1328955)

The authors consider the Cauchy problem for a system of partial differential equations describing the evolution of turbulent bursts in the framework of the \(b\)-\(\varepsilon\) model: \(b_ t= ((b^ 2/ \varepsilon) b_ x)_ x- \varepsilon\), \(\varepsilon_ t= ((b^ 2/ \varepsilon) \varepsilon_ x )_ x- \gamma (\varepsilon^ 2/b)\); \(x\in \mathbb{R}\), \(t>0\); \(b(x,0)= b_ 0(x)\), \(\varepsilon (x,0)= \varepsilon_ 0(x)\), \(\gamma\) is a positive parameter. In the present paper, the existence theory is extended to the new case \(0<\gamma <1\), and the large time behaviour of the solution is investigated for \(0<\gamma \leq 3/2\).