Global regularity of the multi-dimensional compressible Navier-Stokes-Korteweg system with large initial data

In this work, we establish the global existence of strong solutions to the 2D and 3D compressible Navier-Stokes-Korteweg system with arbitrarily large initial data on the torus. This system was derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985] and is widely used to model capillarity in compressible fluids. Via an original modified Nash-Moser type iteration, we establish a critical novel estimate linking the effective velocity and the lower bound of the density, which plays a crucial role in deriving the positive lower bound of the density. To our knowledge, this can be viewed as the first existence result of global strong solutions for the compressible fluid dynamics equations with physical significance in general three-dimensional domains with arbitrarily large initial data.