Blow-up solutions for the Constantin-Lax-Majda equation with a generalized viscosity term.

Abstract

A generalized one-dimensional model for the three-dimensional vorticity equation of incompressible and viscous fluid is considered. Its viscosity term is given by an arbitrary order of derivative of the vorticity. A formal solution of the equation is given explicitly by using the spectral method. We investigate convergence of the solution and show that the solution blows up in finite time for sufficiently small viscosity coefficient regardless of the order of derivative of the viscosity term.

Conclusion

We investigated a generalized one-dimensional model for the three-dimensional vorticity equation of incompressible and viscous flows. The viscosity term is given by an arbitrary order derivative of the vorticity. Whatever order derivative of the vorticity the viscosity term has, we proved mathematically that the solution of the equation blows up in finite time for a small range of the viscosity coefficient if all the initial Fourier coefficients are non-negative. For other range of initial data, we show some numerical examples that indicate blow-up of the solution.

Reference

T. Sakajo Blow-up solutions of Constantin-Lax-Majda equation with a generalized viscosity term, J. Math. Sci. Univ. Tokyo , vol. 10 (2003) pp. 187-207.

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