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Licensed Unlicensed Requires Authentication Published by Princeton University Press 2024

1.4 Linear theory

From the book Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik

  • Maria Colombo , Vikram Giri , Maximilian Janisch , HyunjuVE Kwon , Elia Brué and Dallas Albritton
https://doi.org/10.1515/9780691257846-009
© 2024 Princeton University Press, Princeton
From the book
Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik
Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik

Chapters in this book (44)

Frontmatter
Contents
Preface
Acknowledgments
Introduction
0.1 Idea of the proof
0.2 Differences with Vishik’s work
0.3 Further remarks
1 General strategy: Background field and self-similar coordinates
1.1 The initial velocity and the force
1.2 The infinitely many solutions
1.3 Logarithmic time scale and main Ansatz
1.4 Linear theory
1.5 Nonlinear theory
1.6 Dependency tree
2 Linear theory: Part I
Introduction
2.1 Preliminaries
2.2 Proof of Theorem 2.4 and proof of Theorem 2.1(a)
2.3 Proof of Theorem 1.10: preliminary lemmas
2.4 Proof of Theorem 1.10: conclusion
3 Linear theory: Part II
Introduction
3.1 Preliminaries
3.2 The eigenvalue equation and the class
3.3 A formal expansion
3.4 Overview of the proof of Theorem 3.12
3.5 ODE Lemmas
3.6 Proof of Proposition 3.13
3.7 Proof of Proposition 3.15: Part I
3.8 Proof of Proposition 3.15: Part II
3.9 Proof of Proposition 3.17
3.10 Proof of Lemma 3.19
4 Nonlinear theory
Introduction
4.1 Proof of Proposition 4.2
4.2 Proof of Lemma 4.3
4.3 Proof of the baseline L2 estimate
4.4 Estimates on the first derivative
Appendix A. A more detailed spectral analysis
A.1 From Remark 3.3(i) to Remark 2.2(c)
A.2 Proof of Remark 3.3(i)
A.3 Proof of Theorem 3.4
A.4 Proof of Proposition A.4
Appendix B. Proofs of technical statements
B.1 Proof of Remark 0.2
B.2 Proof of Theorem 0.3
B.3 Proof of Proposition 1.5
B.4 Proof of Lemma 1.9
Bibliography
Index
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