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Singularity formation for complex solutions of the 3D incompressible Euler equations. (English) Zbl 0789.76013
Summary: Moore’s approximation method [D. W. Moore, Proc. R. Soc. London, Ser. A 365, 105-119 (1979; Zbl 0404.76040)], first formulated for vortex sheets, is generalized and applied to axi-symmetric flow with swirl and with smooth initial data. The approximation preserves the forward cascade of energy but neglects any backflow of energy. It splits the Euler equations into two sets of equations: one for \(u_ +=u_ +(r,z,t)\) containing all non-negative wavenumbers (in \(z\)) and the second for \(u_ -=\bar u_ +\). The equations for \(u_ +\) are exactly the Euler equations but with complex initial data. Traveling waves solutions \(u_ +=u_ +(r,z-i\sigma t)\) with imaginary wave speed are found numerically for this problem. The asymptotic properties of the resulting Fourier coefficients show a singularity forming in finite time at which the velocity blows up.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
Software:
MPFUN
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