Optimal analyticity estimates for non-linear active-dissipative evolution equations

Abstract

Active-dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterised by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative), and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth order dissipation, and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalising the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilising a criterion involving the rate of growth of suitable norms of the $n$-th spatial derivative of the solution, with respect to the spatial variable, as $n$ tends to infinity. An estimate of the rate of growth of the $n$-th spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if $\gamma $, the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e., if $\gamma $ is not larger that one, then the solution is not in general analytic. Extensive numerical experiments are undertaken to confirm the analysis and also to compute the band of analyticity of solutions for a wide range of active-dissipative terms and large spatial periods that support chaotic solutions. These ideas can be applied to a wide class of active-dissipative-dispersive pseudo-differential equations.