Abstract:

We survey some old and more recent results regarding evolution of the energy $e$ ($L^2$-norm) and enstrophy $E$ ($H^1$-norm) of the solutions to the 2D space periodic Navier-Stokes equations driven by a time-independent external force $f$. It turns out that the long-time dynamics is dominated by the parabolic scaling $E^2/e$. Indeed, evolution of this quantity shows that the long-time dynamics of the solutions lies inside the region $E^2\le e$. Interestingly, the solutions can approach boundary of this region — the parabola $E^2=|f|_{L^2}^2e$ — only at the origin or at the points of intersection of this parabola with the “spectral rays” $E=\lambda e$, where $\lambda$ is an eigenvalue of the Stokes operator. In the latter case the external forcing $f$ must itself be an eigenvector of the Stokes operator corresponding to the eigenvalue $\lambda$ and the intersection of $E^2=|f|_{L^2}^2e$. These intersections correspond to the stationary solutions $u_\lambda=f/\nu\lambda$.