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The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in SL (2 , R). In the case of perfect fluids, the motion is given by geodesic flow in SL (2 , R) with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in SL (2 , R). A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as t→ ± ∞. For MHD, solutions are bounded and generically quasi-periodic and recurrent.

We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C γ ρ γ for γ > 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied.