Global large solutions and incompressible limit for the compressible Navier-Stokes system with capillarity

Consider the Cauchy problem for the barotropic compressible Navier-Stokes-Korteweg equations in the whole space Rd (d > 2), supplemented with large initial velocity v0 and almost constant initial density p0. In the two-dimensional case, the global solutions are shown in the critical Besov spaces framework without any restrictions on the size of the initial velocity, provided that the pressure admits a stability condition and the volume viscosity is sufficiently large. The result still holds for the higher dimensional case d > 3 under the additional assumption that the classical incompressible Navier-Stokes equations, supplemented with the initial velocity as the Helmholtz projection of v0, admits a global strong solution.(c) 2022 Elsevier Inc. All rights reserved.