Asymptotics Toward Rarefaction Waves and Vacuum for 1-d Compressible Navier-Stokes Equations

In this work we study time asymptotic behavior of solutions of 1-d Compressible Navier-Stokes equations toward solutions of Euler equations that consist of two rarefaction waves separated by a vacuum region. The analysis relies on various energy estimates. 0.1 Summary and the main result Time asymptotic toward rarefaction waves for Compressible Navier-Stokes equations for barotropic fluids was studied in [2, 3]. It was proved that if V (x/t), U (x/t) is a solution of Euler equations  vt = ux, ut = − p(v)x, p = (v)−γ, γ ≥ 1, (t, x) ∈ R × R (1) with Riemann initial data (vr, ur) = { (V , U), x > 0, (V −, U−), x < 0, which consists of one or two rarefaction waves and such that sup s V (s) <∞, (2) then, the solution v, u of Compressible Navier-Stokes equations  vt = ux, ut − ( 1 vux)x + p(v)x = 0, p(v) = v, γ ≥ 1, (t, x) ∈ R × R, (3) with the initial data (v0, u0) such that lim x→±∞ (v0(x), u0(x)) = (V ±, U±),