108 research outputs found
Dynamics of symplectic fluids and point vortices
We present the Hamiltonian formalism for the Euler equation of symplectic
fluids, introduce symplectic vorticity, and study related invariants. In
particular, this allows one to extend D.Ebin's long-time existence result for
geodesics on the symplectomorphism group to metrics not necessarily compatible
with the symplectic structure. We also study the dynamics of symplectic point
vortices, describe their symmetry groups and integrability.Comment: 12 page
Asymptotic directions, Monge-Ampere equations and the geometry of diffeomorphism groups
In this note we obtain the characterization for asymptotic directions on
various subgroups of the diffeomorphism group. We give a simple proof of
non-existence of such directions for area-preserving diffeomorphisms of closed
surfaces of non-zero curvature. Finally, we exhibit the common origin of the
Monge-Ampere equations in 2D fluid dynamics and mass transport.Comment: 10 pages, 1 fig., to appear in J. of Math. Fluid Mechanic
Shock waves for the Burgers equation and curvatures of diffeomorphism groups
We establish a simple relation between 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 establishes an intrinsic connection between ideal Euler
hydrodynamics (via Arnold's approach), shock formation in the multidimensional
Burgers equation and the Wasserstein geometry of the space of densities.Comment: 11 pages, 2 figure
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