Analyticity of dissipative-dispersive systems in higher dimensions

TL;DR

This paper proves that solutions to certain higher-dimensional dissipative-dispersive equations are analytic when the dissipation order exceeds one, extending previous techniques to a broader class of equations with universal attractors.

Contribution

It extends the analyticity results to higher-dimensional equations with universal attractors, including the Topper–Kawahara and Frenkel–Indireshkumar equations, for dissipation order greater than one.

Findings

Solutions are analytic when dissipation order > 1

Techniques from previous work are adapted for higher dimensions

Numerical evidence supports the optimality of the dissipation threshold

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.