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Weak Solutions of the Euler Equations and Onsager’s Conjecture

Haghshenas, Mahdi | 2021

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 54944 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Safdari, Mohammad
  7. Abstract:
  8. The Onsager's conjecture is concerned with the dichotomy between rigidity and flexibility of weak solutions of incompressible Euler equations. Lars Onsager conjectured that weak solutions of Euler equations that are not smooth enough could be dissipative, even without the help of viscosity. On the other hand, it is well known that $C^1$ solutions conserve energy. Onsager conjectured that C^(1/3) regularity marks the threshold for this dichotomy. In other words, Hölder continuous solutions with Hölder exponent greater than 1/3 conserve the energy, while for every Hölder exponent less than 1/3, there are dissipative Hölder continuous solutions. The threshold 1/3 is intimately tied with Kolmogorov's theory for turbulence. The first part of the conjecture was proven in 1994. Then the second part was achieved in 2018 using the new perspective on Euler equations and convex integration techniques. Besides the proof of Onsager's conjecture, convex integration has also been used to obtain several other results in this field, such as the nonuniqueness of weak solutions of Euler and Navier-Stokes equations. In this thesis, we review the properties of Euler and Navier-Stokes equations, and we present the Onsager's conjecture and its motivations. Then we explain the convex integration techniques along with the proof of the Onsager's conjecture.
  9. Keywords:
  10. Euler Equation ; Navier-Stokes Equation ; Onsager Model ; H-Principle ; Anomalous Dissipation ; Convex Integration

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