A class of global large solutions to the compressible Navier–Stokes–Korteweg system in critical Besov spaces

Abstract

In this paper, we consider the Cauchy problem for the compressible Navier–Stokes–Korteweg system in critical Besov spaces. The global solutions are established under a nonlinear smallness assumption on the initial data, whether the sound speed is positive or equal to zero. Furthermore, we explain that this kind of nonlinear smallness condition is large in the sense that we can construct an example of initial data satisfying it, even though each component of the initial velocity can be arbitrarily large in $$\dot{B}^{\frac{n}{p}-1}_{p,1}$$.