Regularity Criteria Of The Boussinesq Equations And The Regularity Estimates Given By The Eigenvalues Of The Deformation Tensor

The two and three dimensional Boussinesq equations are discussed in this paper.In the first part,we give regularity estimates for local regular solutions of the initial-boundary-value-problem with periodic boundary condition for the Boussinesq equations.This follows light on the possibility of the solution blowing-up in finite time.IL the 2D case,an L2 estimates for the temperature gradient is given in terms of the eigenvalues of the deformation tensor.It follows from this estimate that if the rate of deformation of a fluid element is large,the regular solution is more likely to blow-up.In the 3D case,an L2 estimate for the vorticity is given in terms of the eigenvalues of the deformation tensor and the derivatives of the temperature.From the estimate,we can see that if for most of the time,all of the fluid elements are stretched planarly,and the temperature gradient is small,then the regular solution is more likely to blow-up.In the contrary,if linear stretching dominates and the temperature gradient is bounded,then the solution is less likely to blow-up.In the second part,two regularity criteria of the Boussinesq equations are given by velocity gradient and the temperature gradient.They are particularly interesting in the absence of viscosity and thermal diffusion.In dimension two,the global regularity problem is open only in this case and our result implies that velocity gradient is equally important for this problem as temperature gradient.In dimension three,regularity criterion for this case is scarce.