  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.
