Structure stabilization of transonic shocks by mass additions for compressible Euler flows in two-dimensional straight nozzles

The structure stability of stationary transonic shocks plays a crucial role in aerodynamics for designing supersonic nozzles. In this paper, we show that mass additions stabilize transonic shocks in steady compressible Euler flows within two-dimensional straight nozzles. The presence of a source term in the equation of mass prevents the widely used method of Lagrange transform or stream functions. Moreover, mass addition leads to background solutions of the system being non-trivial functions without smallness of C1 norm. To overcome these difficulties, we propose a novel decomposition for the subsonic Euler equations that effectively separates the elliptic and hyperbolic modes. The new approach necessitates solving three coupled sub-problems: (i) a Dirichlet–Neumann–Venttsel-mixed boundary value problem of a second-order elliptic equation for pressure, with the equation containing various integral and pointwise nonlocal terms; (ii) an initial boundary value problem of transport equations for entropy and total enthalpy; (iii) a family of two-point boundary value problems of ordinary differential equations on tangential velocity in each cross-section of the nozzle. This approach allows us to clarify the mathematical mechanisms that promote shock stability. We establish the structural stability of almost every background solution under small perturbations of the inlet supersonic flows and outlet pressures. Furthermore, all the physical quantities in our solution possess the same regularity. This new approach may be applicable to solve other free boundary problems associated with nonlinear systems of conservation laws of elliptic–hyperbolic composite-mixed type.