On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term.
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 3-D Euler equation. It is shown by - that
the solution of the model equation blows up in finite time for sufficiently small viscosity,
however large diffusion term it may has. In the present article, we discuss the existence of a unique
global solution for large viscosity.
Conclusion
We consider the global existence of solutions for the Constantin-Lax-Majda equation with the generalized viscosity term, which is proposed as a one-dimensional model for the 3-D Navier-Stokes equations. When the order of diffusion term $\alpha > 1$, the solution, whose initial Fourier coefficients are non-negative, exists globally in time for sufficiently large viscosity coefficient. On the other hand, when $\alpha <0$ there exists no global solution no matter how large the viscosity coefficient is. It means that the quadratic term $H(\omega)\omega$ has such strong nonlinearity that weak
diffusion is impossible to control its growth.
In 3-D Navier-Stokes equations, the solution exists globally in time regardless of the viscosity
coefficient when $\alpha > \frac{5}{4}$. Hence, the present global result and the blow-up result by S-, which showed that the solution of the CLM equation blew up in finite time in $L^2$
for all order of diffusion term if the viscosity coefficient was sufficiently small, indicate that
the CLM equation with the generalized viscosity term fails to
catch this analytic property of the 3-D Navier-Stokes equations.
Reference
T. Sakajo On global solutions of Constantin-Lax-Majda equation with a generalized viscosity term, Nonlinearity , vol. 16 (2003) pp. 1319-1328.
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