  This paper is to study global-in-time existence of weak solutions to zero
Mach number system which derives from the full Navier-Stokes system, under a
special relationship between the viscosity coefficient and the heat
conductivity coefficient such that, roughly speaking, 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 in \cite{Danchin-Liao},
for arbitrary large initial data, we will show that this unique strong solution
exists globally in time, by a weak-strong uniqueness argument.
