Stability of Transonic Contact Discontinuity for Two-Dimensional Steady Compressible Euler Flows in a Finitely Long Nozzle

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

“…Let Ω(h) V I be the triangle bounded by l 1 − , l 1 + and the lower nozzle wall Γ(h) − . Finally, we remark that different from the argument made in [36], we need to consider the Hölder estimates, i.e., C 1,α -norm on the charateristic curves and the solutions instead of the C 1 -norm used in [36].…”

“…III is similar. First, we can follow the approach in [36] for proving the Proposition 4.1 and apply the Lemma 4.2 to get…”

“…For other related problems, one can see [6,11,13,46] for the sonic-subsonic limit, [12] for the incompressible limit, [24][25][26]42] for the jet flow, [28] for the axisymmetric flow with nontrivial swirl, and [30] for the subsonic flow in a finite nozzle. For the supersonic flow in nozzles, one basic feature is that a shock will be generated if the nozzles are taking sufficiently long (see [36,Appendix]). We see that most of these results are on the nozzles with special structures, for instance, the supersonic flow through an expanding nozzle, which was proposed by Courant-Friedrichs in [27] and was studied recently in [22,47,52].…”

“…For the supersonic flow, the stability over a 2D Lipschitz wall in the BV space was proved in [15]. Recently, the well-posedness theory for the steady supersonic compressible Euler flows through a 2D finitely long nozzle with a contact discontinuity was established in [36]. For other related problems on the steady supersonic contact discontinuity over a wedge with a sharp convex corner, we refer the reader to [14,29,37,[42][43][44]48].…”

“…The transonic contact discontinuity is a free boundary that separates the subsonic flow on the upper layer and supersonic flow on the lower layer in the nozzle. The length of the nozzle is taken to be finite since the singularities will generally develop from the smooth supersonic flow when the nozzle is infinitely long (see the Appendix of [36]). Moreover, the loss of the regularity for the supersonic flow will lead to the low regularities of the boundary as well as the data along the boundary for the subsonic region, which makes the elliptic boundary value problem for the subsonic flow difficult to deal with.…”