We consider the problem of convergence of sorutions of the one-dimensional compressible Navier-Stokes equations to a solution of the associate Lagrange system when the viscosity and heat conductivity tend to zero. Under appropriate entropy assumptions, using compensated compactness techniques, we obtain a peculiar kind of convergence, which allows us to pass to the limit in the non-linear flux functions.
We consider the following system of equations for a viscous heat conducting polytropic gas in Lagrangean coordinates in one space dimension:
(1p"
where c,B = e = p v / ( yl), e being the internal' energy, 8 the temperature, y the adiabatic exponent and c, the specific heat coefficient.
The existence for (l)"." with initial-boundary value data:
Cu(x, O),v(x, 01, P(X, 011 = Cuo(x), uo(x), P,(X)l, X E CO,lI,
[u,(x), po(x)] > 0, has been proved in [6]. We deal with the problem of the convergence of solutions of (l)"." to a solution of where the data (i) are smooth and miq,, the Euler system v, -u, = 0, u, + P x = 0, (e +id), + ( u p ) , = 0, when E, K + 0.
We assume the following a priori bounds on the solutions of (1 '*I().