Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation (Q1978107)

The author considers the Kuramoto-Sivashinsky equation \(u_t+u_{xxxx}+u_{xx}+uu_x=0\), where \(u=u(x,t)\) is \(L-\)periodic in \(x\), \(x\in \mathbb{R}\), \(t>0\), \(\int_0^Lu(x,t)dx =0\). The phase space of the equation is the Hilbert space. The main statement is that the radius of special analyticity on the global attractor is a lower-semicontinuous function at the stationary solutions. The existence is proved of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter \(L\) is proved. As an application of the main result the author proves that the number of rapid spatial oscillations of functions belonging to this neighborhood is, up to a logarithmic correction, at most linear in \(L\).