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On a generalization of the Constantin-Lax-Majda equation. (English) Zbl 1221.35300
The authors consider a slightly generalised form of one-dimensional De Gregorio’s equation that describes vorticity dynamics in a three-dimensional incompressible viscous flow. The equation involves velocity and its gardient as coefficients and the velocity gradient is determined by the Hilbert transform of the vorticity. One obtains Constantin-Lax-Majda’s equation if the convection term is discarded for which it is shown that most of the solutions blow up in finite time. A theorem which yields a mathematical criterion to test whether a global solution exists has been proven. This criterion requires that the time integral on a finite interval of the norm of the Hilbert transform of the vorticity in an appropriate Lebesgue space of infinite order should be finite. The authors are unable to prove mathematically that this criterion is satisfied for the equation under consideration. However, numerical computation indicates that the probability is high for global solution to exist.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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