The Low Mach Number Limit of Global Solutions to the Full Compressible Navier–Stokes System in Critical Besov Spaces With Large Initial Data

We are concerned with global existence of regular solutions to full compressible Navier–Stokes equations and their asymptotic behavior when the Mach number is sufficiently small. We establish global existence in critical Besov spaces for arbitrary large initial data provided that the divergence‐free component of initial velocity and the difference between initial temperature and density generate a global regular solution to incompressible Boussinesq systems. Moreover, we rigorously justify the convergence to the incompressible model as the Mach number tends to zero. The proof relies on a fine‐grained analysis of the high–middle–low frequencies of density, velocity, and temperature. Our result can be seen as an improvement on Danchin and He [Mathematische Annalen 366 no. 3‐4 (2016): 1365–1402], including the extension from small initial data to large initial data and new convergence results which hold at the level of critical regularity.