Lower bound estimates for viscosity solutions to isentropic gas dynamics and to Euler equations

The main contribution of this paper is to use the maximum principle to obtain a lower bound ρϵ⩾C(t,ϵ)>0 for the parabolic system generated by adding “artificial viscosity” to the system of one-dimensional isentropic gas dynamics and to the Euler equations corresponding to the Broadwell model [R.E. Caflisch, Navier–Stokes and Boltzmann shock profiles for a model of gas dynamics, Comm. Pure Appl. Math. 32 (1979) 521–554], where C(t,ϵ) are given explicitly. Here we do not need the restrictions and the periodic initial data used in previous work by other authors.