M. F. Atiyah and I. M. Singer, The index of elliptic operators, IV. Ann. of Math, vol.93, issue.2, pp.119-138, 1971.

T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften, vol.252

. Springer-verlag, , 1982.

M. Bauer, M. Bruveris, P. Harms, and P. W. Michor, Smooth perturbations of the functional calculus and applications to riemannian geometry on spaces of metrics, 2018.

M. Bauer, M. Bruveris, and B. Kolev, Fractional Sobolev metrics on spaces of immersed curves, Calc. Var. Partial Differential Equations, vol.57, issue.1, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01486557

M. Bauer, J. Escher, and B. Kolev, Local and Global Well-posedness of the fractional order EPDiff equation on R d, Journal of Differential Equations, vol.258, issue.6, pp.2010-2053, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01111245

M. Bauer, P. Harms, and P. W. Michor, Sobolev metrics on shape space of surfaces, J. Geom. Mech, vol.3, issue.4, pp.389-438, 2011.

M. Bauer, P. Harms, and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics, J. Differential Geom, vol.94, issue.2, pp.187-208, 2013.

M. Bauer, P. Harms, and P. W. Michor, Fractional sobolev metrics on spaces of immersions, 2019.

M. Bauer, P. Harms, and S. C. Preston, Vanishing distance phenomena and the geometric approach to SQG, Archive for Rational Mechanics and Analysis, 2018.

M. Bauer, B. Kolev, and S. C. Preston, Geometric investigations of a vorticity model equation, J. Differential Equations, vol.260, issue.1, pp.478-516, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01203765

M. Bauer and K. Modin, Semi-invariant riemannian metrics in hydrodynamics, 2018.

Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc, vol.2, pp.225-255, 1989.

Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math, vol.52, pp.411-452, 1999.

M. Bruveris, Regularity of maps between sobolev spaces, vol.52, pp.11-24, 2017.

M. Bruveris, The L 2 -metric on C ? (M, N ). arXiv e-prints, 2018.

M. Bruveris and F. Vialard, On completeness of groups of diffeomorphisms, Journal of the European Mathematical Society, vol.19, issue.5, pp.1507-1544, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01253261

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett, vol.71, issue.11, pp.1661-1664, 1993.

J. Chemin, Équations d'Euler d'un fluide incompressible, Facettes mathématiques de la mécanique des fluides, pp.9-30, 2010.

E. Cismas, Euler-Poincaré-Arnold equations on semi-direct products II, Discrete Contin. Dyn. Syst, vol.36, issue.11, pp.5993-6022, 2016.

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv, vol.78, issue.4, pp.787-804, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00003261

P. Constantin, P. D. Lax, and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math, vol.38, issue.6, pp.715-724, 1985.

P. Constantin, A. J. Majda, and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, vol.7, issue.6, p.1495, 1994.

D. G. Ebin, The manifold of Riemannian metrics, Global Analysis (Proc. Sympos, vol.XV, pp.11-40, 1968.

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

D. G. Ebin, J. E. Marsden, and A. E. Fischer, Diffeomorphism groups, hydrodynamics and relativity, Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress Differential Geometry and Applications, vol.1, pp.135-279, 1971.

Y. V. Egorov and M. A. Shubin, Partial Differential Equations VI: Elliptic and Parabolic Operators, 1990.

J. Eichhorn, Global analysis on open manifolds, 2007.

J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech, vol.6, issue.3, pp.335-372, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00673137

J. Escher, B. Kolev, and M. Wunsch, The geometry of a vorticity model equation, Commun. Pure Appl. Anal, vol.11, issue.4, pp.1407-1419, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00528726

L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable. Mémoires de l'académie des sciences de Berlin, vol.14, pp.154-193, 1765.

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. J, vol.36, issue.3, pp.323-344, 1989.

F. Gay-balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Bra?ov Ser. III, vol.2, issue.51, pp.55-58, 2009.

N. Grosse and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr, vol.286, issue.16, pp.1586-1613, 2013.

G. Grubb, Distributions and operators, Graduate Texts in Mathematics, vol.252, 2009.

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, The breadth of symplectic and Poisson geometry, vol.232, pp.203-235, 2005.

L. Hörmander, The analysis of linear partial differential operators, III. Classics in Mathematics, 2007.

H. Inci, T. Kappeler, and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol.226, 2013.

B. Khesin and G. Misio, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math, vol.176, issue.1, pp.116-144, 2003.

B. Khesin and V. Ovsienko, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen, vol.21, issue.4, pp.81-82, 1987.

B. Khesin and R. Wendt, The geometry of infinite-dimensional groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol.51

. Springer-verlag, , 2009.

B. Kolev, Local well-posedness of the EPDiff equation: a survey, The Journal of Geometric Mechanics, vol.9, issue.2, pp.167-189, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01397112

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys, vol.40, issue.2, pp.857-868, 1999.

S. T. Melo, T. Schick, and E. Schrohe, Families index for Boutet de Monvel operators, Münster J. Math, vol.6, pp.343-364, 2013.

P. Michor and D. Mumford, On Euler's equation and 'EPDiff, The Journal of Geometric Mechanics, vol.5, issue.3, pp.319-344, 2013.

J. Milnor, Remarks on infinite-dimensional Lie groups, Relativity, groups and topology, pp.1007-1057, 1983.

G. Misio, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys, vol.24, issue.3, pp.203-208, 1998.

G. Misio and S. C. Preston, Fredholm properties of Riemannian exponential maps on diffeomorphism groups, Invent. Math, vol.179, issue.1, pp.191-227, 2010.

K. Modin, Generalized hunter-saxton equations, optimal information transport, and factorization of diffeomorphisms, The Journal of Geometric Analysis, vol.25, issue.2, pp.1306-1334, 2014.

H. Omori, On the group of diffeomorphisms on a compact manifold, Global Analysis (Proc. Sympos, vol.XV, pp.167-183, 1968.

H. Omori, Translated from the 1979 Japanese original and revised by the author, Translations of Mathematical Monographs. American Mathematical Society, vol.158, 1997.

M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries, Pseudo-Differential Operators. Theory and Applications, vol.2, 2010.

S. Shkoller, Geometry and curvature of diffeomorphism groups with H 1 metric and mean hydrodynamics, J. Funct. Anal, vol.160, issue.1, pp.337-365, 1998.

S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom, vol.55, issue.1, pp.145-191, 2000.

A. Shnirelman, Generalized fluid flows, their approximation and applications, Geometric and Functional Analysis, vol.4, pp.586-620, 1994.

A. I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid. Mat. Sb, vol.144, pp.82-109, 1985.

F. Tiglay and C. Vizman, Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys, vol.97, issue.1, pp.45-60, 2011.

H. Triebel, Theory of Function Spaces II, 1992.

A. Trouvé and L. Younes, Local geometry of deformable templates, SIAM J. Math. Anal, vol.37, issue.1, pp.17-59, 2005.

P. Washabaugh, The SQG equation as a geodesic equation, Arch. Ration. Mech. Anal, vol.222, issue.3, pp.1269-1284, 2016.

M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric, J. Nonlinear Math. Phys, vol.17, issue.1, pp.7-11, 2010.