Global Large Solutions to the Three Dimensional Compressible Navier--Stokes Equations

This paper is devoted to the global large solutions to the 3 dimensional compressible Navier--Stokes equations in the critical Besov spaces with initial data satisfying a nonlinear smallness condition. Here the “large solutions" mean that the any component of the initial velocity could be arbitrarily large. Moreover, we give an example of initial data satisfying the nonlinear smallness condition, while the norms of each component are arbitrarily large. Our approach is inspired by the weighted Chemin--Lerner technique used for the incompressible Navier--Stokes equations.