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Vortex dynamics in \(\mathbb{R}^4\). (English) Zbl 1273.76069
Summary: The vortex dynamics of Euler’s equations for a constant density fluid flow in \(\mathbb{R}^4\) is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form \(\omega\) in \(\mathbb{R}^4\). These distributions are supported on two-dimensional surfaces termed membranes and are the analogs of vortex filaments in \(\mathbb{R}^3\) and point vortices in \(\mathbb{R}^2\). The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation. The regularized self-induced velocity field is then shown to be proportional to the mean curvature vector field of the membrane but rotated by \(90^{\circ}\) in the plane of normals. Next, the Hamiltonian membrane model is presented. The symplectic structure for this model is derived from a general formula for vorticity distributions due to J. Marsden and A. Weinstein [Physica D 7, 305–323 (1983; Zbl 0576.58008)]. Finally, the dynamics of the four-form \(\omega\) is examined. It is shown that Ertel’s vorticity theorem in \(\mathbb{R}^3\), for the constant density case, can be viewed as a special case of the dynamics of this four-form.
©2012 American Institute of Physics

MSC:
76B47 Vortex flows for incompressible inviscid fluids
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