Sharp well-posedness and spatial decaying for a generalized dispersive-dissipative Kuramoto-type equation and applications to related models

TL;DR

This paper introduces a general dispersive-dissipative nonlinear equation with fractional Laplacians, analyzing its well-posedness and spatial decay properties, which are relevant for physical fluid models and solitary wave behavior.

Contribution

It establishes sharp well-posedness results and optimal decay estimates for solutions of a broad class of fractional dispersive-dissipative equations, including several physically relevant models.

Findings

Proved well-posedness in Sobolev spaces of negative order.

Derived optimal spatial decay properties of solutions.

Connected decay results with physical models and numerical observations.

Abstract

We introduce a fairly general dispersive-dissipative nonlinear equation, which is characterized by fractional Laplacian operators in both the dispersive and dissipative terms. This equation includes some physically relevant models of fluid dynamics as particular cases. Among them are the \emph{dispersive Kuramoto-Velarde}, the \emph{Kuramoto-Sivashinsky} equation, and some nonlocal perturbations of the \emph{KdV} and the \emph{Benjamin-Ono} equations. We thoroughly study the effects of the fractional Laplacian operators in the qualitative study of solutions: on the one hand, we prove a sharp well-posedness result in the framework of Sobolev spaces of negative order, and on the other hand, we investigate the pointwise decaying properties of solutions in the spatial variable, which are optimal in some cases. These last results are of particular interest for the corresponding physical models. Precisely, they align with previous numerical works on the spatial decay of a particular kind of solutions, commonly referred to as solitary waves.