We consider non-linear Schrödinger equations with small complex coefficient of size 6 in front of the Laplacian. The space-variable belongs to the unit n-cube (n = 3) and Dirichlet boundary conditions are assumed on the cube's boundary. The equations are studied in the turbulent regime which means that d « 1 and supremum-norms of the solutions we consider are at least of order one. We prove that space-scales of the solutions are bounded from below and from above by some finite positive degrees of d and show that this result implies non-trivial restrictions on spectra of the solutions, related to the Kolmogorov-Obukhov five-thirds law (these restrictions are less specific than the 5/3-law, but they apply to a much wider class of solutions). Our approach is rather general and is applicable to many other nonlinear PDEs in the turbulent regime. Unfortunately, it does not apply to the Navier-Stokes equations.
|Number of pages
|Geometric and Functional Analysis
|Published - 1999