Global small solution and optimal decay rate for the Korteweg system in Besov spaces

TL;DR

This paper proves the global existence and optimal decay rates of strong solutions for the Korteweg system with small initial data in Besov spaces, extending understanding of its long-term behavior in higher dimensions.

Contribution

It establishes the global well-posedness and decay rates for the Korteweg system in Besov spaces, using Friedrich method and compactness, for the first time in this setting.

Findings

Global well-posedness for small initial data

Optimal decay rates in Besov spaces

Extension to general pressure in higher dimensions

Abstract

In this paper we consider the Cauchy problem to the Korteweg system with the general pressure in dimension $d\geq2$, and establish the global well-posedness of strong solution for the small initial data in $L^p$ type critical Besov spaces by using the Friedrich method and compactness arguments. Furthermore, we also obtain the optimal decay rate for the Korteweg system in $L^2$ type Besov spaces.