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Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions

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Abstract

We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss–Codazzi equations through examples for the Euler equations for fluids and the Euler–Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics.

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Author information

Authors and Affiliations

  1. Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Amit Acharya

  2. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Gui-Qiang G. Chen & Siran Li

  3. AMSS & UCAS, Chinese Academy of Sciences, Beijing, 100190, 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

    Dehua Wang

Authors
  1. Amit Acharya
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  2. Gui-Qiang G. Chen
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  3. Siran Li
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  4. Marshall Slemrod
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  5. Dehua Wang
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Correspondence to Gui-Qiang G. Chen.

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Communicated by D. Kinderlehrer

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Acharya, A., Chen, GQ.G., Li, S. et al. Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions. Arch Rational Mech Anal 226, 1009–1060 (2017). https://doi.org/10.1007/s00205-017-1149-5

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  • DOI: https://doi.org/10.1007/s00205-017-1149-5

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