Spatial decay properties for a model in shear flows posed on the cylinder

Abstract

We study spatial decay properties for solutions of the Pelinovski-Stepanyants equation posed on the cylinder. We establish the maximum polynomial decay admissible for solutions of such a model. It is verified that the equation on the cylinder propagates polynomial weights with different restrictions than the model set in $\mathbb{R}^2$. For example, a local well-posedness theory is deduced which contains the line solitary wave provided by solutions of the Benjamin-Ono equation extended to the cylinder. Key results for our analysis are obtained from a detailed study of the dispersive effects of the linear equation in weighted Sobolev spaces. The results in this paper appear to be one of the first studies of polynomial decay for nonlocal models in the cylinder.