The three‐dimensional incompressible Navier–Stokes equations with the hyperdissipation false(−normalΔfalse)γ always possess global smooth solutions when γ≥54. Tao [6] and Barbato, Morandin and Romito [1] made logarithmic reductions in the dissipation and still obtained the global regularity. This paper makes a different type of reduction in the dissipation and proves the global existence and uniqueness in the H1‐functional setting.
The three-dimensional axisymmetric Euler equations without swirl can be represented by the conservation of ! Â =r along the particle trajectory, where ! Â denotes the swirl component of the vorticity. The two-dimensional Euler equation shares a parallel representation. Delort's work has long settled the global existence of weak solutions corresponding to a vortex sheet data of distinguished sign. In contrast, the parallel global existence problem for the axisymmetric Euler equations remains an outstanding open problem. This paper establishes the global existence of weak solutions to the axisymmetric Euler equations without swirl when the initial vorticity ! Â 0 obeys ! Â 0 =r 2 L 1 R 3 \ L p R 3 for p 2 .1, 1/. The approach is the method of viscous approximations. A major step in the proof is to extract a strongly convergent subsequence of solutions to a viscous approximation of the Euler equations.
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