Spectral Properties of Solutions for Nonlinear PDEs in the Turbulent Regime

Abstract.

We consider non-linear Schrödinger equations with small complex coefficient of size $\delta$ in front of the Laplacian. The space-variable belongs to the unit n-cube $(n \le {\bf 3}$ and Dirichlet boundary conditions are assumed on the cube's boundary. The equations are studied in the turbulent regime which means that $\delta \ll 1$ 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 $\delta$ 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.