Analyticity for the Kuramoto-Sivashinsky equation

Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

“…Setting A = 0 in the above and noting that in one dimension (7) produces a bound on J" h 2 depending only on T, one can directly reproduce the well known global existence result [20,21,32,37,46,47] for the one-dimensional Kuramoto-Sivashinsky equation.…”

Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

“…possesses a finite-dimensional global attractor in L 2 (0, L), and [5] that this attractor is bounded in the Gevrey class G ρ for some ρ > 0. (For the restricted case of odd initial conditions such results were obtained previously -the existence of an attractor by Nicolaenko et al [48] and the Gevrey regularity by Liu [43]; some further details on Gevrey regularity for solutions on the attractor are given by Grujić [29].)…”

An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation

“…There have been numerous analytical [15][16][17][18][19] and numerical [20][21][22][23] studies of the 1D KSE with 2L-periodic boundary conditions (see below also). The 1D KSE has been proved to possess a unique smooth solution that depends continuously on its initial data [19].…”