Global in Time Stability of Steady Shocks in Nozzles

We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem. 1. Steady shocks in channels The inviscid compressible isentropic Euler equations in dimension d = 1 are ρt + (ρu)x = 0, (ρu)t + (ρu 2 + p(ρ))x = 0 , The first expresses conservation of mass ρ and the second conservation of momentum ρu. The system is strictly hyperbolic when ∀ρ > 0, p(ρ) > 0 , p′(ρ) > 0, p′′(ρ) ≥ 0 . The sound speed is defined to be c := pρ(ρ) 1/2 . The local speeds are u ± c. When |u| > c the flow is supersonic and where |u| < c it is subsonic. We treat 1d channel or nozzle flow with u > 0. Transverse averaged 1d nozzle with section a(x) yields the system ρt + (ρu)x = − a′(x) a(x) ρu, (ρu)t + (ρu 2 + p(ρ))x = − a′(x) a(x) ρu . This has the conservative form (a(x)ρ)t + (a(x)ρu)x = 0, (a(x)ρu)t + (a(x)ρu 2 + p(ρ))x = 0 . Suppose the channel flat outside 0 ≤ x ≤ 1, that is, a′ ∈ C∞ 0 (]0, 1[). We study the stability of steady transonic shocks. The flow is supersonic to the left and subsonic to the right. The steady hypothesis is crucial. It allows us to truncate the Séminaire Laurent-Schwartz — EDP et applications Centre de mathématiques Laurent Schwartz, 2011-2012 Exposé no II, 1-11