The Euler equations as a differential inclusion

Abstract

We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

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      }
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      EDITION = {third},
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      FPUBLISHER = {North-Holland Publishing Co.},
      ADDRESS = {Amsterdam},
      YEAR = {1984},
      PAGES = {xii+526},
      ISBN = {0-444-87558-1; 0-444-87559-X},
      MRCLASS = {76-02 (35Q10 65-02 76-08 76D05)},
      MRNUMBER = {86m:76003},
      ZBLNUMBER = {0568.35002},
      }
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      author = {De Lellis, Camillo and Széklyhidi Jr., Lászlo},
      TITLE = {On admissibility criteria for weak solutions of the Euler equations},
      TYPE = {preprint},
      ARXIV = {0712.3288},
      YEAR = {2007},
      }

Authors

Camillo De Lellis

Institut für Mathematik
Universität Zürich
CH-8057 Zürich
Switzerland

László Székelyhidi Jr.

Departement Mathematik
ETH Zürich
CH-8092 Zürich
Switzerland