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Transonic Shocks in Multidimensional Divergent Nozzles

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Abstract

We establish existence, uniqueness and stability of transonic shocks for a steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity (non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Fréchet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Myoungjean Bae

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706-1388, USA

    Mikhail Feldman

Authors
  1. Myoungjean Bae
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  2. Mikhail Feldman
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Correspondence to Myoungjean Bae.

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Communicated by C. Dafermos

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Bae, M., Feldman, M. Transonic Shocks in Multidimensional Divergent Nozzles. Arch Rational Mech Anal 201, 777–840 (2011). https://doi.org/10.1007/s00205-011-0424-0

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  • DOI: https://doi.org/10.1007/s00205-011-0424-0

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