Archive for Rational Mechanics and Analysis

, Volume 225, Issue 1, pp 141–176 | Cite as

Global Existence and Asymptotic Behavior of Affine Motion of 3D Ideal Fluids Surrounded by Vacuum

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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 of 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.

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA

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