2013/12/03 by Ciprian Foias, Michael S. Jolly, Foias, Ciprian +9
Mathematics · #35B41 #35Q30 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:35B41 #msc:35Q30
paper · pdf · doi:10.48550/arxiv.1312.0929
40 pages
arxiv created 2013/12/03 · arxiv updated 2013/12/04
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∈\calD(A(α-1)/(2)), we show there is a fixed strip about the real time axis on which a uniform bound |Aαu|< mανκ0α holds for each α∈ \bN. Here ν is viscosity, \k0=2π/L, and mα is explicitly given in terms of g and α. We show that if any element in \calA is in \D(Aα), then all of \calA is in \D(Aα), and likewise with \D(Aα) replaced by C^∞(Ω). 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.