形变张量的特征值与Boussinesq方程组的正则性估计

讨论了二维及三维满足周期边界条件的Boussinesq方程初边值问题的局部正则解在有限时间内爆破的可能性.在二维情况下，用形变张量的特征值给出温度梯度的L2估计，从中看出若流体微团变形的速率大，则解爆破的可能性就大.在三维情况下，用形变张量的特征值和温度的偏导给出涡量的L2估计，从中发现若流体微团在大部分时间内一般是平面拉伸，且温度的偏导较小时，解爆破的可能性就大；若一般是线性拉伸，温度的偏导又不任意增大时，解爆破的可能性就小.

Eigenvalues of the Deformation Tensor and Regularity Estimates for the Boussinesq Equations

The blow-up possibility of local regular solutions to the initial-boundary-value problems with periodic boundary conditions for 2D and 3D Boussinesq systems was discussed.In the 2D case,an L2 estimate of the temperature gradient was given in terms of the eigenvalues of the deformation tensor.From this estimate it is found that if the deformation rate of a fluid element is large,the regular solution is more likely to blow up.In the 3D case,an L2 estimate of the vorticity was given in terms of the eigenvalues of the deformation tensor and the derivatives of temperature.From this estimate it is shown that if for most of the time,most of the fluid elements are stretched in plane and the temperature gradient is small,the regular solution is more likely to blow up.On the contrary,if linear stretching dominates and the temperature gradient is bounded,the solution is less likely to blow up.