2020/12/18 by Feimin Huang, Huang, Feimin, Jie Kuang +5
Mathematics · #35B07 #35B20 #35D30 #76J20 #76L99 #76N10 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B07 #msc:35B20 #msc:35D30 #msc:76J20 #msc:76L99 #msc:76N10
paper · pdf · doi:10.48550/arxiv.2012.10039
arxiv created 2021/08/29 · arxiv updated 2021/08/31
We consider the stability of transonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle. This is the first work on the mixed-type problem of transonic flows across a contact discontinuity as a free boundary in nozzles. We start with the Euler-Lagrangian transformation to straighten the contact discontinuity in the new coordinates. However, the upper nozzle wall in the subsonic region depending on the mass flux becomes a free boundary after the transformation. Then we develop new ideas and techniques to solve the free-boundary problem in three steps: (1) we fix the free boundary and generate a new iteration scheme to solve the corresponding fixed boundary value problem of the hyperbolic-elliptic mixed type by building some powerful estimates for both the first-order hyperbolic equation and a second-order nonlinear elliptic equation in a Lipschitz domain; (2) we update the new free boundary by constructing a mapping that has a fixed point; (3) we establish via the inverse Lagrangian coordinates transformation that the original free interface problem admits a unique piecewise smooth transonic solution near the background state, which consists of a smooth subsonic flow and a smooth supersonic flow with a contact discontinuity.