Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum. (English) Zbl 1367.35115

Summary: 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 \(\mathrm{GL}^+(3, \mathbb{R})\). 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 \(\mathrm{SL}(3, \mathbb{R})\) 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.


35Q31 Euler equations
76N15 Gas dynamics (general theory)
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
53D25 Geodesic flows in symplectic geometry and contact geometry
35R35 Free boundary problems for PDEs
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