From Isometric Embeddings to Turbulence

László Székelyhidi

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Submission date: 20. Dec. 2012
Pages: 54
published in: HCDTE lecture notes. Part II. Nonlinear hyperbolic PDEs, dispersive and transport equations / G. Alberti ... (eds.)
Springfield : American Institute of Mathematical Sciences, 2013. - P. 195 - 255
(AIMS series on applied mathematics ; 7) 
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Abstract:
The following dichotomy concerning isometric embeddings of the sphere is well-known: whereas the only C2 isometric embedding of S2 into R3 is the standard embedding modulo rigid motion, there exists many C1 isometric embeddings which can "wrinkle" S2 into arbitrary small regions. The latter flexibility, known as the Nash-Kuiper theorem, involves an iteration scheme called convex integration which turned out to have surprisingly wide applicability. These lecture notes are meant as an analysts exposition of convex integration. In particular the notes contain:

1. an essentially complete proof of the rigidity and flexibility for isometric embeddings,
2. a general discussion of the existence theory for first order partial differential inclusions and
3. the recent application of the same methods to the incompressible Euler equations of fluid mechanics, extending the work of Scheffer and Snirelman.