L. Ambrosio and N. Gigli, A User???s Guide to Optimal Transport, Lecture Notes in Mathematics, 2013.
DOI : 10.1007/978-3-642-32160-3_1

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.635.5738

V. Arnold, Sur la g??om??trie diff??rentielle des groupes de Lie de dimension infinie et ses applications ?? l'hydrodynamique des fluides parfaits, Annales de l???institut Fourier, vol.16, issue.1, pp.319-361, 1966.
DOI : 10.5802/aif.233

J. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numerische Mathematik, vol.84, issue.3, pp.375-393, 2000.
DOI : 10.1007/s002110050002

Y. Brenier, The dual least action problem for an ideal, incompressible fluid Archive for Rational Mechanics and Analysis, pp.323-351, 1993.

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, vol.117, issue.4, pp.375-417, 1991.
DOI : 10.1002/cpa.3160440402

Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Communications on Pure and Applied Mathematics, vol.111, issue.4, pp.411-452, 1999.
DOI : 10.1002/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3

Y. Brenier, Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics, Calculus of Variations and Partial Differential Equations, vol.4, issue.2, pp.55-64, 2013.
DOI : 10.1007/s00526-012-0510-7

URL : https://hal.archives-ouvertes.fr/hal-00827679

A. Bressan and A. Constantin, Global Conservative Solutions of the Camassa???Holm Equation, Archive for Rational Mechanics and Analysis, vol.27, issue.5, pp.215-239, 2007.
DOI : 10.1007/s00205-006-0010-z

A. Bressan and M. Fonte, An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation, Methods and Applications of Analysis, vol.12, issue.2, pp.191-219, 2005.
DOI : 10.4310/MAA.2005.v12.n2.a7

D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, 2001.
DOI : 10.1090/gsm/033

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, vol.71, issue.11, pp.1661-1664, 1993.
DOI : 10.1103/PhysRevLett.71.1661

URL : http://arxiv.org/abs/patt-sol/9305002

L. Chizat, G. Peyré, B. Schmitzer, and F. Vialard, Unbalanced Optimal Transport: Geometry and Kantorovich Formulation. ArXiv e-prints, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01271981

L. Chizat, B. Schmitzer, G. Peyré, and F. Vialard, An Interpolating Distance Between Optimal Transport and Fisher???Rao Metrics, Foundations of Computational Mathematics, vol.5, issue.2, 2016.
DOI : 10.1007/s10208-016-9331-y

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.457, issue.2008, pp.953-970, 2001.
DOI : 10.1098/rspa.2000.0701

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, vol.181, issue.2, pp.229-243, 1998.
DOI : 10.1007/BF02392586

A. Constantin and D. Lannes, The hydrodynamical relevance of the camassa?holm and degasperis?procesi equations. Archive for Rational Mechanics and Analysis, pp.165-186, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00281511

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, vol.14, issue.8, pp.953-988, 2001.

G. , D. Philippis, and A. Figalli, The Monge-Ampère equation and its link to optimal transportation, Bull. Amer. Math. Soc, vol.51, pp.527-580, 2014.

G. David, J. Ebin, and . Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math, vol.92, issue.2, pp.102-163, 1970.

J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, Journal of Geometric Mechanics, vol.6, issue.3, pp.335-372, 2014.
DOI : 10.3934/jgm.2014.6.335

URL : https://hal.archives-ouvertes.fr/hal-00673137

J. Escher and B. Kolev, The Degasperis???Procesi equation as a non-metric Euler equation, Mathematische Zeitschrift, vol.46, issue.3, pp.1137-1153, 2011.
DOI : 10.1007/s00209-010-0778-2

URL : https://hal.archives-ouvertes.fr/hal-00409022

D. S. Freed and D. Groisser, The basic geometry of the manifold of riemannian metrics and of its quotient by the diffeomorphism group. Michigan Math, 23] S. Gallot. ´ Equations différentielles caractéristiques de la sphère. Annales scientifiques de l'Ecole Normale Superieure, pp.323-344235, 1979.

T. O. Gallouët and L. Monsaingeon, A JKO Splitting Scheme for Kantorovich--Fisher--Rao Gradient Flows, SIAM Journal on Mathematical Analysis, vol.49, issue.2, 2016.
DOI : 10.1137/16M106666X

F. Gay-balmaz, C. Tronci, and C. Vizman, Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups, Journal of Geometric Mechanics, vol.5, pp.39-84, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01092293

K. Grunert, H. Holden, and X. Raynaud, Lipschitz metric for the periodic Camassa???Holm equation, Journal of Differential Equations, vol.250, issue.3, pp.1460-1492, 2011.
DOI : 10.1016/j.jde.2010.07.006

D. D. Holm, J. E. Marsden, and T. S. Ratiu, The Euler???Poincar?? Equations and Semidirect Products with Applications to Continuum Theories, Advances in Mathematics, vol.137, issue.1, pp.1-81, 1998.
DOI : 10.1006/aima.1998.1721

URL : http://doi.org/10.1006/aima.1998.1721

B. Khesin, J. Lenells, G. Misiolek, and S. C. Preston, Curvatures of Sobolev metrics on diffeomorphism groups, Pure and Applied Mathematics Quarterly, vol.9, issue.2, pp.291-332, 2013.
DOI : 10.4310/PAMQ.2013.v9.n2.a2

URL : http://arxiv.org/abs/1109.1816

B. Khesin, J. Lenells, G. Misio, and S. C. Preston, Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics, Geometric and Functional Analysis, vol.9, issue.1, pp.334-366, 2013.
DOI : 10.1007/s00039-013-0210-2

B. Khesin and R. Wendt, The geometry of infinite-dimensional groups & Business Media, 2008.

I. Kolá?, P. W. Michor, and J. Slovák, Natural operations in differential geometry, 1993.
DOI : 10.1007/978-3-662-02950-3

S. Kondratyev, L. Monsaingeon, and D. Vorotnikov, A new optimal trasnport distance on the space of finite Radon measures, Adv. Differential Equations, vol.21, issue.11, pp.1117-1164, 2016.

S. Kouranbaeva, The Camassa???Holm equation as a geodesic flow on the diffeomorphism group, Journal of Mathematical Physics, vol.40, issue.2, pp.857-868, 1999.
DOI : 10.1063/1.532690

S. Lang, Fundamentals of differential geometry, volume 191 of Graduate Texts in Mathematics, 1999.

J. Lenells, Conservation laws of the Camassa???Holm equation, Journal of Physics A: Mathematical and General, vol.38, issue.4, pp.869-880, 2005.
DOI : 10.1088/0305-4470/38/4/007

M. Liero, A. Mielke, and G. Savaré, Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures. ArXiv e-prints, 2015.
DOI : 10.1137/15m1041420

URL : http://arxiv.org/abs/1509.00068

M. Liero, A. Mielke, and G. Savaré, Optimal Transport in Competition with Reaction: The Hellinger--Kantorovich Distance and Geodesic Curves, SIAM Journal on Mathematical Analysis, vol.48, issue.4, pp.2869-2911, 2016.
DOI : 10.1137/15M1041420

J. Lott, Some Geometric Calculations on Wasserstein Space, Communications in Mathematical Physics, vol.26, issue.2, pp.423-437, 2008.
DOI : 10.1007/s00220-007-0367-3

URL : http://arxiv.org/abs/math/0612562

J. Maas, M. Rumpf, C. Schönlieb, and S. Simon, A generalized model for optimal transport of images including dissipation and density modulation, ESAIM: Mathematical Modelling and Numerical Analysis, vol.49, issue.6, 2015.
DOI : 10.1051/m2an/2015043

R. J. Mccann, Polar factorization of maps on Riemannian manifolds, Geometric and Functional Analysis, vol.11, issue.3, pp.589-608, 2001.
DOI : 10.1007/PL00001679

P. Henry and . Mckean, Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math, vol.57, issue.3, pp.416-418, 2004.

Q. Mérigot and J. Mirebeau, Minimal geodesics along volume preserving maps, through semi-discrete optimal transport. ArXiv e-prints, 2015.

M. Micheli, P. W. Michor, and D. Mumford, Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds, Izvestiya: Mathematics, vol.77, issue.3, p.541, 2013.
DOI : 10.1070/IM2013v077n03ABEH002648

URL : https://hal.archives-ouvertes.fr/hal-01444168

P. W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol.93, 2008.
DOI : 10.1090/gsm/093

W. Peter, D. Michor, and . Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math, vol.10, pp.217-245, 2005.

G. Misiolek, Classical solutions of the periodic camassa?holm equation. Geometric & Functional Analysis GAFA, pp.1080-1104, 2002.

G. Misiolek, . Stephenc, and . Preston, Fredholm properties of Riemannian exponential maps on??diffeomorphism groups, Inventiones mathematicae, vol.179, issue.1, pp.191-227, 2010.
DOI : 10.1007/s00222-009-0217-3

K. Modin, Generalized Hunter???Saxton Equations, Optimal Information Transport, and Factorization of Diffeomorphisms, The Journal of Geometric Analysis, vol.4, issue.6, pp.1306-1334, 2015.
DOI : 10.1007/s12220-014-9469-2

J. Moser, On the volume elements on a manifold, Transactions of the American Mathematical Society, vol.120, issue.2, pp.286-294, 1965.
DOI : 10.1090/S0002-9947-1965-0182927-5

F. Otto, THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION, Communications in Partial Differential Equations, vol.4, issue.1-2, pp.101-174, 2001.
DOI : 10.1007/BF00535689

F. Rezakhanlou, Optimal transport problems for contact structures, 2015.

R. T. Rockafellar, Integrals which are convex functionals, Pacific Journal of Mathematics, vol.24, issue.3, pp.439-469, 1971.
DOI : 10.2140/pjm.1968.24.525

URL : http://projecteuclid.org/download/pdf_1/euclid.pjm/1102986512

A. Trouvé and L. Younes, Metamorphoses Through Lie Group Action, Foundations of Computational Mathematics, vol.5, issue.2, pp.173-198, 2005.
DOI : 10.1007/s10208-004-0128-z

C. Villani, Topics in optimal transportation. Number 58, 2003.
DOI : 10.1090/gsm/058

C. Villani, Optimal transport: old and new Springer Science & Business Media, CMLS, vol.338, 2008.
DOI : 10.1007/978-3-540-71050-9