Transonic flows with shocks past curved wedges for the full Eulerequations

Gui-Qiang  Chen; Jun  Chen; Mikhail  Feldman

doi:10.3934/dcds.2016.36.4179

Article Contents

2016, Volume 36, Issue 8: 4179-4211. Doi: 10.3934/dcds.2016.36.4179

This issue Previous Article Low-dimensional Galerkin approximations of nonlinear delay differential equations Next Article The unsteady transonic small disturbance equation: Data on oblique curves

Transonic flows with shocks past curved wedges for the full Euler equations

1.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radclie Observatory Quarter, Woodstock Road, Oxford, OX2 6GG
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055
3.
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1388

Received: April 2015

Revised: January 2016

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the transformation from Eulerian to Lagrangian coordinates and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for the nonlinear equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the shock location or equivalently the entropy function along the shock-front downstream, all the physical variables can be expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at infinity can be uniquely determined with a uniform decay rate. To solve the free boundary problem, we employ the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the nonlinear equation, and then update the entropy function to prove that the updating map has a fixed point. Another advantage in our analysis is in the context of the full Euler equations so that the Bernoulli constant is allowed to change for different fluid trajectories.

Keywords:

Transonic flow,
full Euler equations,
transonic shock,
curved wedge,
supersonic,
subsonic,
shock-front,
mixed type,
mixed-composite hyperbolic-elliptic type,
free boundary problem,
existence,
stability,
asymptotic behavior,
decay rate.

Mathematics Subject Classification: 35M12, 35R35, 76H05, 76L05, 35L67, 35L65, 35B35, 35Q31, 76N10, 76N15, 35B30, 35B40, 35Q35.

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