Analyticity of the attractors of dissipative‐dispersive systems in higher dimensions

Abstract

We investigate the analyticity of the attractors of a class of Kuramoto‐Sivashinsky–type pseudodifferential 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 ∇nu, as n tends to infinity (here, u is the solution). These techniques can now be utilized 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 γ, the order of dissipation of the pseudodifferential operator, is higher than one. We believe that this estimate is optimal based on numerical evidence.