Global strong solutions to the $3$D rotating compressible Navier--Stokes--Korteweg system for large data in the critical $\widehat{L^p}$ framework

Let us consider the $3$D compressible Navier–Stokes–Korteweg system in the rotational framework. Although there is a wealth of literature on the weak solutions to this system, there seem to be no results on the strong solutions. In this paper, we show the unique existence of global solutions for {\it large} initial data in the critical Besov-type spaces based on the Fourier–Lebesgue spaces $\widehat{L^p}({\mathbb{R}}^3)$ with $2\le p<3$, provided that the rotation speed and the Mach number are sufficiently large and small, respectively. The key ingredient of the proof is to establish the Strichartz-type estimates due to the dispersion caused by the mixture of the rotation and acoustic waves in the Fourier–Lebesgue spaces, and focus on the better structure of dissipation from the Korteweg term and the nonlinear terms of the divergence form in the momentum formulation.