Optimal transport, Euler equations, Mather and DiPerna-Lions theories

Contents
Introduction 7
1 The optimal transport problem 9
1.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Weak regularity of the optimal transport map . . . . . . . . . . . . . . . . 11
1.2.2 Displacement convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Sub-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Regularity of the optimal transport on Riemannian manifolds . . . . . . . . . . . 16
1.5 The (anisotropic) isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . 18
1.5.1 Stability of isoperimetric problems . . . . . . . . . . . . . . . . . . . . . . 21
1.5.2 An isoperimetric-type inequality on constant curvature manifolds . . . . . 23
1.6 The optimal partial transport problem . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Variational methods for the Euler equations 27
2.1 Arnorld’s interpretation and Brenier’s relaxation . . . . . . . . . . . . . . . . . . 27
2.2 A study of generalized solutions in 2 dimensions . . . . . . . . . . . . . . . . . . 29
2.3 A second relaxed model and the optimality conditions . . . . . . . . . . . . . . . 30
3 Mather quotient and Sard Theorem 35
3.1 The dimension of the Mather quotient . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 The connection with Sard Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 A Sard Theorem in Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 DiPerna-Lions theory for non-smooth ODEs 41
4.1 A review of DiPerna-Lions and Ambrosio’s theory . . . . . . . . . . . . . . . . . 41
4.2 The stochastic extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 The infinite dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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