Analyticity of dissipative-dispersive systems in higher dimensions

Abstract

We investigate the analyticity of the attractors of a class of Kuramoto-Sivashinsky type pseudo-differential equations in higher dimensions, which are periodic in all spatial variables and possess a universal attractor. This is done by fine-tuning the techniques used in a previous work of the second author, which are based on an analytic extensibility criterion involving the growth of $\nabla^n u$, as $n$ tends to infinity (here $u$ is the solution). These techniques can now be utilised in a variety of higher dimensional equations possessing universal attractors, including Topper--Kawahara equation, Frenkel--Indireshkumar equations and their dispersively modified analogs. We prove that the solutions are analytic whenever $\gamma$, the order of dissipation of the pseudo-differential operator, is higher than one. We believe that this estimate is optimal, based on numerical evidence.