Asymptotic stability of transonic steady nozzle flows
In the present article, we prove the nonlinear asymptotic stability of stationary solutions to the quasi-one-dimensional barotropic Euler equations in a finite-length nozzle with variable cross-sectional area $A(x)$. We consider all stationary flow configurations arising from subsonic inflow: flows that remain subsonic throughout the nozzle, flows that pass continuously through a sonic state and remain supersonic up to the exit, and flows that subsequently return to the subsonic regime through a stationary shock. We prove that all these stationary solutions are asymptotically stable with respect to sufficiently small perturbations. No smallness restrictions are imposed on the mass flux, the magnitude of $A_x$, or the prescribed exit pressure, provided that the corresponding stationary solution exists. The only geometrical assumptions are that $A_{xx}>0$ at a smooth subsonic--supersonic transition, and $A_x>0$ at a stationary supersonic--subsonic shock. To the best of our knowledge, this gives the first nonlinear asymptotic stability result for a stationary flow passing smoothly from the subsonic to the supersonic regime in a finite-length nozzle. The principal analytical difficulty is the degeneracy of the characteristic structure at the sonic point. We overcome this difficulty by coupling energy structures adapted respectively to the subsonic and supersonic regimes. This coupling provides the dissipation needed to control the smooth sonic transition, and when combined with the dynamics of the shock, yields a unified nonlinear stability framework for smooth and discontinuous transonic nozzle flows.