Abstract
We consider a one-dimensional model for the three-dimensional vorticity equation of incompressible and viscous fluids. This model is obtained by adding a generalized viscous diffusion term to the Constantin–Lax–Majda equation, which was introduced as a model for the three-dimensional Euler equation (Constantin P, Lax P D and Majda A 1985 A simple one-dimensional model for the three-dimensional vorticity equation Commun. Pure. Appl. Math. 38 715–24). It is shown in Sakajo T (2003 Blow-up solutions of the Constantin–Lax–Majda equation with a generalized viscosity term J. Math. Sci. Univ. Tokyo 10 187–207) that the solution of the model equation blows up in finite time for sufficiently small viscosity, however large a diffusion term it may have. In this paper, we discuss the existence of a unique global solution for large viscosity.
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