Determining Free Boundary in a Nozzle for Compressible Euler Flow

This project aims to develop rigorous mechanisms and mathematical theories on determining the free boundary in a finitely long nozzle for multi-dimensional steady compressible Euler flow by studying several important problems. In particular, it mainly focuses on the location of the subsonic contact discontinuity with a given pressure at the outlet of the nozzle, the location of the transonic shock with combustion phenomenon in a finitely long nozzle, and other related problems. Starting from the seminal works in [6, 7, 42], there are many results on the steady gas flow with a free boundary in nozzles, for example, transonic shocks in nozzles [3, 4, 8, 12–15, 20–24, 28–34, 38, 39, 41, 43] or contact discontinuities in a finitely long nozzle [25–27]. Following [18], we will impose the receiver pressure condition at the exit of the nozzle, the slip boundary condition along the nozzle walls, and appropriate boundary conditions at the inlet of the nozzle depending on whether the incoming flow is supersonic or subsonic. Mathematically, these problems can be formulated into a free boundary value problem governed by nonlinear equations. The key difficulty is that the free boundary can be arbitrarily shifted for uniform flows. Therefore, finding plausible mechanisms to determine the free boundary location in a horizontal flat nozzle is a longstanding mathematical open problem. Recently for the transonic shock in a horizontal flat nozzle problem, two mechanisms have been introduced based on the nonflatness of the nozzle wall in [24] and the non-zero swirl velocity in [22]. However, rigorous mathematical analysis for such problems is far from being satisfied. We believe the ideas and techniques developed in this project would be helpful for the mathematical theory of the multi-dimensional compressible Euler equations and nonlinear partial differential equations. We have been working on the related problems for several years. We find a mechanism based on non-zero swirl velocity to determine the location of the transonic normal shock in [22], and establish the existence and uniqueness of subsonic contact discontinuity in an infinitely long nozzle with large vorticity in [10]; the stability of supersonic/transonic contact discontinuity in a finitely long nozzle in [25-27]; stability of three-dimensional attached transonic weak shock in [5]; and convexity of self-similar transonic shocks in [9]. In this project, we will mainly focus on further developing the techniques used in the works above, to study the mentioned objectives.