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 [2]. It is shown in [6] 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.

Finite time singularities for a class of generalized surface quasi-geostrophic equations

“…We remark that (1.2) can be regarded as a special case of our model (1.1) if one chooses β = 0 and θ 0 (x 1 , x 2 ) = θ 0 (x 1 ). Some other 1D models, all of which have some analogy with the 2D SQG and the 3D Euler equation in vorticity form, can be found in [2], [9], [6], [18], [24], [25] and [26].…”

Well-posedness for a transport equation with nonlocal velocity

“…For other results about various 1D models, we refer the readers to Baker, Li and Morlet [1], Dong, Du and Li [12], Kiselev, Nazarov and Shterenberg [14], Morlet [16], Sakajo [18], Schochet [19], Wegert and Vasudeva Murthy [20] and references therein.…”

Profile for a Simultaneously Blowing up Solution to a Complex Valued Semilinear Heat Equation

“…Equation (1) has a strong relation with the viscous Constantin-Lax-Majda equation, which is a one dimensional model for the vorticity equation. For more details see Okamoto, Sakajo and Wunsch [OSW08], Sakajo [Sak03a] and [Sak03b] and Guo, Ninomiya, Shimojo and Yanagida in [GNSY13].…”