Low mach Number Limit of the Viscous and Heat Conductive Flow with general pressure law on torus

We prove the low Mach number limit from compressible Navier-Stokes-Fourier system with the general pressure law around a constant state on the torus $\mathbb{T}^N_a$. We view this limit as a special case of the weakly nonlinear-dissipative approximation of the general hyperbolic-parabolic system with entropy. In particular, we consider the ill-prepared initial data, for which the group of fast acoustic waves is needed to be filtered. This extends the previous works, in particular Danchin [ Amer. J. Math. 124 (2002), 1153-1219] in two ways: 1. We treat the fully general non-isentropic flow, i.e. the pressure depends on the density $\rho$ and temperature $\theta$ by basic thermodynamic law. We illustrate the role played by the entropy structure of the system in the coupling of the acoustic waves and incompressible flow, and the construction of the filtering group. 2. We refine the small divisor estimate, which helps us to give the first explicit convergence rate of the filtered acoustic waves whose propogation is governed by non-local averaged system. In previous works, only convergence rate of incompressible limit was obtained.