The 3D compressible and incompressible Euler equations with a physical vacuum
free boundary condition and affine initial conditions reduce to a globally
solvable Hamiltonian system of ordinary differential equations for the
deformation gradient in $\rm{GL}^+(3,\mathbb R)$. The evolution of the fluid
domain is described by a family ellipsoids whose diameter grows at a rate
proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit
of the fluid ellipsoid is determined by a positive semi-definite quadratic form
of rank $r=1$, 2, or 3, corresponding to the asymptotic degeneration of the
ellipsoid along $3-r$ of its principal axes. In the compressible case, the
asymptotic limit has rank $r=3$, and asymptotic completeness holds, when the
adiabatic index $\gamma$ satisfies $4/3<\gamma<2$. The number of possible
degeneracies, $3-r$, increases with the value of the adiabatic index $\gamma$.
In the incompressible case, affine motion reduces to geodesic flow in
$\rm{SL}(3,\mathbb R)$ with the Euclidean metric. For incompressible affine
swirling flow, there is a structural instability. Generically, when the
vorticity is nonzero, the domains degenerate along only one axis, but the
physical vacuum boundary condition fails over a finite time interval. The
rescaled fluid domains of irrotational motion can collapse along two axes.
