Global existence of solutions of an extended Landau equation (Q1370986)

The author studies an equation arising in the theory of hydro-dynamic stability which is a generalization of the Landau equation. It models the amplitude of a disturbance of a boundary layer as a function of time. This equation is obtained by reducing a certain system of nonstationary Navier-Stokes equations with the aid of an asymptotic analysis of spatially periodic solutions. The amplitude of a disturbance of small but finite size is written as a function of the time \(t\) in the form \(\varepsilon\cdot q(t)\) with a certain parameter \(\varepsilon>0\), \(q(t)\geq 0\) and \(q(t)\) being of order 1 as \(\varepsilon\to 0\). \(\varepsilon\) can be identified with \(\sqrt{|\omega|}\) and \(q(t)\) satisfies the equation \[ q'(t)=2\varepsilon^2 q(t)(\delta+f[q](t)),\quad t>0\text{ and } q(0)=q_0 \tag{1} \] where \(\delta=\pm 1\) is identical with the sign of \(\omega\) (\(\omega\) is the temporal amplification rate of a spatially periodic disturbance of infinitesimal small size of a boundary layer) and \(f[q]\) denotes a certain functional of \(q\). The main result (Theorem 1) guarantees the existence of a continuously differentiable global solution of (1) (with \(\delta=1\)) which has a limit at infinity for sufficiently small \(\varepsilon>0\) (independently of \(q_0\)). The proof is based on the fixed point principle. The Landau equation can be obtained from (1) by setting \(f[q]=\Lambda q\) (\(\Lambda\) being a constant).