Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum

TL;DR

This paper analyzes the global existence and long-term behavior of affine motions in 3D ideal fluids with vacuum boundaries, revealing how domain shapes evolve and degenerate over time under different physical conditions.

Contribution

It provides a comprehensive analysis of affine solutions to 3D Euler equations with vacuum boundaries, including asymptotic behavior and stability properties, extending previous understanding of fluid domain evolution.

Findings

Ellipsoidal domains grow linearly over time.

Asymptotic limits depend on the adiabatic index and can degenerate along axes.

Incompressible affine flows reduce to geodesic flows in SL(3,R).

Abstract

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.