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

TL;DR

This paper investigates the low Mach and Froude number limits for classical solutions of a 1-D compressible Navier-Stokes free boundary problem, establishing uniform estimates and convergence to steady states without small initial data assumptions.

Contribution

It provides the first analysis of low Mach and Froude limits for free boundary problems, including uniform estimates and convergence rates for all-time classical solutions.

Findings

Uniform estimates for solutions with respect to Mach and Froude numbers

Convergence of solutions to steady state as parameters vanish or time tends to infinity

Establishment of all-time existence of classical solutions with sharp convergence rates

Abstract

In this paper, we study the low Mach and Froude number limit for the all-time classical solution of a fluid-vacuum free boundary problem of one-dimensional compressible Navier-Stokes equations. No smallness of initial data for the existence of all-time solutions are supposed. The uniform estimates of solutions with respect to the Mach number and the Froude number are established for 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 is interesting 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, the steady state. The main difficulty is that, the system is degenerate near the free boundary and contains singular terms. This result can be viewed as the first 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 classical solution with sharp convergent rates to the steady state, while previous results are only concerned with the weak or strong solutions.