Time analyticity with higher norm estimates for the 2D Navier-Stokes equations

Abstract

This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on $[0,L{]}^2$, and a force $g\in\calD(A^{\frac{α-1}{2}})$, we show there is a fixed strip about the real time axis on which a uniform bound $|A^{\alpha }u|<m_{\alpha }\nu {\kappa }_0^{\alpha }$ holds for each $α\in \bN$. Here $\nu$ is viscosity, $\k0=2π/L$, and $m_{\alpha }$ is explicitly given in terms of $g$ and $\alpha$. We show that if any element in $\calA$ is in $\D(A^α)$, then all of $\calA$ is in $\D(A^α)$, and likewise with $\D(A^α)$ replaced by $C^{\infty }(\Omega )$. We demonstrate the universality of this "all for one, one for all" law on the union of a hierarchal set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a nonzero force by showing that if this is so, the force must be in a particular function class.

40 pages

Cited by in corpus (2)

• One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations
• On whether zero is in the global attractor of the 2D Navier-Stokes equations