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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.

MSC:

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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