Rigorous numerics for dissipative partial differential equations. II: Periodic orbit for the Kuramoto-Sivashinsky PDE -- a computer-assisted proof
[For part I see \textit{P. Zgliczynśki} and \textit{K. Mischaikow}, ibid. 1, No. 3, 255--288 (2001; Zbl 0984.65101)]. A method of self-consistent a priori bounds for dissipative partial differential equations is developed. These bounds allow studying rigorously the dynamic of dissipative partial differential equations. As an application of the theory a computer-assisted proof of the existence of a periodic orbit for the Kuramoto-Sivashinsky equation \(u_t= (u^2)_x- u_{xx}- vu_{xxxx}\), \(u(t,x)= u(t,x+ 2\pi)\), \(u(t,x)= -u(t, -x)\), where \(v= 0,127\) is presented. After the introduction the base of the method of self-consistent a priori bounds, which sets up the framework for treating all high-dimensional Galerkin projections as a single differential inclusion is presented. In section 3 the author discusses the notion of a Poincaré section and Poincaré return map in the context of self-consistent bounds. In section 4 as an application a result about the existence of periodic orbit of the Kuramoto-Sivashinsky equation is presented. In the section 5 the description of computer-assisted proof of the existence is described. In the last section a modification of the Lohner algorithm is presented, which was used in the rigorous integration of a differential inclusion in the proof of the existence of the orbit.
dissipative partial differential equation
scientific article; zbMATH DE number 2111601