On turbulence in nonlinear Schrödinger equations (Q1380494)

The author considers the small-dispersion and small-diffusion nonlinear Schrödinger equation of the form \[ -iu= -\delta_1\Delta u- i\delta_2\Delta u+| u|^2u+ \zeta_\omega(t,x),\tag{1} \] \(1\geq \delta:= \sqrt{\delta^2_1+ \delta^2_2}> 0\), where the space-variable \(x\) belongs to the unit \(n\)-cube \((n\leq 3)\) and \(u\) satisfies Dirichlet boundary conditions. Moreover, the author assumes that the force \(\zeta\) is a zero-meanvalue random field smooth in \(x\) and stationary in \(t\) with decaying correlations. He proves that the \(C^m\)-norms in \(x\) with \(m\geq 3\) of solutions \(u\), averaged in ensemble and locally averaged in time, are larger than \(\delta^{-km}\), \(k\approx 1/5\). This means that the length-scale of a solution \(u\) decays with \(\delta\) as its positive degree (at least, as \(\delta^k\)), and in some sense, the author proves existence of turbulence for (1).