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In this paper, we study the low Mach and Froude number limit for the all-timeclassical solution of a fluid-vacuum free boundary problem of one-dimensionalcompressible Navier-Stokes equations. No smallness of initial data for theexistence of all-time solutions are supposed. The uniform estimates ofsolutions with respect to the Mach number and the Froude number are establishedfor all the time, in particular for high order derivatives of the pressure,which is a novelty in contrast to previous results. The cases of "ill-prepared"initial data and "well-prepared" initial data are both discussed. It isinteresting to see, either both the Mach number and the Froude number vanish,or the time goes to infinity, the limiting functions are the same, that is, thesteady state. The main difficulty is that, the system is degenerate near thefree boundary and contains singular terms. This result can be viewed as thefirst one on the low Mach and Froude numbers limit for free boundary problems.At the same time, we also establish the all-time existence of the classicalsolution with sharp convergent rates to the steady state, while previousresults are only concerned with the weak or strong solutions.

Low Mach and low Froude number limit for vacuum free boundary problem of all-time classical solutions of 1-D compressible Navier-Stokes equations