Geodesic
Shock Waves for the Burgers Equation and Curvatures of Diffeomorphism Groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 77-85

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We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler hydrodynamics (via Arnold's approach) to shock formation in the multidimensional Burgers equation and the Kantorovich–Wasserstein geometry of the space of densities. 
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     author = {B. A. Khesin and G. Misio{\l}ek},
     title = {Shock {Waves} for the {Burgers} {Equation} and {Curvatures} of {Diffeomorphism} {Groups}},
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B. A. Khesin; G. Misiołek. Shock Waves for the Burgers Equation and Curvatures of Diffeomorphism Groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 77-85. http://geodesic.mathdoc.fr/item/TM_2007_259_a5/