Analyticity for Kuramoto–Sivashinsky‐type equations in two spatial dimensions

Abstract

In this work, we investigate the analyticity properties of solutions of Kuramoto-Sivashinsky-type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three-dimensional models of a spectral method, which was developed by the authors for the one-dimensional Kuramoto-Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function u 2 C 1 , involving the rate of growth of r n u, in suitable norms, as n tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper-Kawahara, Frenkel-Indireshkumar, and Coward-Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors.

Analyticity for a class of nonlocal Kuramoto‐Sivashinsky equations arising in interfacial electrohydrodynamics

“…In the case of Equation 10, where 2 spatial variables appear, the method used to prove the analyticity for this equation is the spectral method, which was developed in Ioakim and Smyrlis. 37 Note that this method, which originally developed in Akrivis et al, 32 was first used to prove analyticity for dissipative-dispersive equations in 1 space dimension. So it remains to investigate if the semigroup method can be used, as the spectral method, to study the analyticity of dissipative-dispersive equations in 2 spatial dimensions.…”

“…for some positive constants c 3 , , and n 0 a sufficiently large positive integer. In Ioakim and Smyrlis, 37 it has been proven the analyticity of solutions of (10) when > 3. The plan of this article is as follows.…”

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

“…In order to prove that h(s) ≤ M(as) s for all s ∈ [0, + − 1], we will make a better estimate of lim sup t→∞ |ŵ k, (t)| rather than (12). It is evident that we may, without loss of generality, assume k, ≥ 0.…”

“…In what follows, we establish the analyticity of the universal attractor of when γ > 1 by fine‐tuning the methods presented in the works of Ioakim and Smyrlis . We present numerical experiments, which suggest that this bound is optimal.…”

“…Demirkaya studied the following two‐dimensional variation of the KS equation with 2 π ‐periodic initial data in both variables, and established the global well‐posedness and the existence of a universal attractor. Ioakim and Smyrlis established the analyticity of the universal attractor of with periodic initial data in two space dimensions provided that γ > 3.…”