(EBOOK PDF) Instability and Non uniqueness for the 2D Euler Equations after M Vishik 1st edition by Camillo De Lellis 9780691257846 0691257841 full chapters

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• ISBN 10:0691257841 

• ISBN 13:9780691257846

• Author:Camillo De Lellis 

Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik
(AMS-219)
An essential companion to M. Vishik’s groundbreaking work in fluid mechanics The incompressible Euler equations are a system of partial differential equations introduced by Leonhard Euler more than 250 years ago to describe the motion of an inviscid incompressible fluid. These equations can be derived from the classical conservations laws of mass and momentum under some very idealized assumptions. While they look simple compared to many other equations of mathematical physics, several fundamental mathematical questions about them are still unanswered. One is under which assumptions it can be rigorously proved that they determine the evolution of the fluid once we know its initial state and the forces acting on it. This book addresses a well-known case of this question in two space dimensions. Following the pioneering ideas of M. Vishik, the authors explain in detail the optimality of a celebrated theorem of V. Yudovich from the 1960s, which states that, in the vorticity formulation, the solution is unique if the initial vorticity and the acting force are bounded. In particular, the authors show that Yudovich’s theorem cannot be generalized to the L^p setting. 

Instability and Non uniqueness for the 2D Euler Equations after M Vishik 1st Table of contents:

 0.1 Idea of the proof
0.2 Differences with Vishik’s work
0.3 Further remarks
Chapter 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
Chapter 2. Linear theory: Part I
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
Chapter 3. Linear theory: Part II
3.1 Preliminaries
3.2 The eigenvalue equation and the class C
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
Chapter 4. Nonlinear theory
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.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
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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Instability and Non,uniqueness for the,Camillo De Lellis