A Global Existence Result for a Zero Mach Number System

In this paper we study the global-in-time existence of weak solutions to a zero Mach number system that derives from the Navier–Stokes–Fourier system, under a special relationship between the viscosity coefficient and the heat conductivity coefficient. Roughly speaking, this relation implies that the source term in the equation for the newly introduced divergence-free velocity vector field vanishes. In dimension two, thanks to a local-in-time existence result of a unique strong solution in critical Besov spaces given by Danchin and Liao (Commun Contemp Math 14:1250022, 2012), for arbitrary large initial data, we show that this unique strong solution exists globally in time, as a consequence of a weak-strong uniqueness argument.