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Nonlinear Conservation Laws and Applications pp 257–266Cite as

Transonic Flows and Isometric Embeddings

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 153))

Abstract

Transonic flows past an obstacle such as an airfoil are first considered. A viscous approximation to the steady transonic flow problem is presented, and its convergence is obtained by the method of compensated compactness. Then the isometric embedding problem in geometry is discussed. A fluid dynamic formulation of the Gauss-Codazzi system for the isometric embedding of two-dimensional Riemannian manifolds is provided, and an existence result of isometric immersions with negative Gauss curvature is given.

AMS(MOS) subject classifications. 76H05, 35M10, 35A35, 76N10, 53C42.

G.-Q. Chen’s research was supported in part by the National Science Foundation under Grants DMS-0807551, DMS-0720925, and DMS-0505473, the Natural Science Foundation of China under Grant NSFC-10728101, and the Royal Society-Wolfson Research Merit Award (UK).

M. Slemrod’s research was supported in part by the National Science Foundation under Grant DMS-0647554.

D. Wang’s research was supported in part by the National Science Foundation under Grants DMS-0604362 and DMS-0906160, and by the Office of Naval Research under Grant N00014-07-1-0668.

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References

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Author information

Authors and Affiliations

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

    Gui-Qiang G. Chen

  2. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK

    Gui-Qiang G. Chen

  3. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Gui-Qiang G. Chen

  4. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Marshall Slemrod

  5. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Marshall Slemrod

  6. Department of Mathematical Sciences, Korean Advanced Institute for Science and Technology, Daejeon, Republic of Korea

    Marshall Slemrod

  7. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Dehua Wang

Authors
  1. Gui-Qiang G. Chen
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  2. Marshall Slemrod
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  3. Dehua Wang
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Editor information

Editors and Affiliations

  1. McAllister Bldg 201, University Park, 16802-6401, Pennsylvania, USA

    Alberto Bressan

  2. , Oxford Centre for Nonlinear PDE, Mathema, University of Oxford, St Giles 24–29, Oxford, OX1 3LB, Montserrat

    Gui-Qiang G. Chen

  3. , School of Mathematics, University of Minnesota, Church Street S.E, 206, Minneapolis, 55455, Minnesota, USA

    Marta Lewicka

  4. , Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260, Pennsylvania, USA

    Dehua Wang

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Chen, GQ.G., Slemrod, M., Wang, D. (2011). Transonic Flows and Isometric Embeddings. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_12

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