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

WANG Zhen; DENG Da-wen

doi:10.21656/1000-0887.370355

Volume 38 Issue 11

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WANG Zhen, DENG Da-wen. Eigenvalues of the Deformation Tensor and Regularity Estimates for the Boussinesq Equations[J]. Applied Mathematics and Mechanics, 2017, 38(11): 1279-1288. doi: 10.21656/1000-0887.370355

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

doi: 10.21656/1000-0887.370355

School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, P.R.China

Received Date: 2016-11-17
Rev Recd Date: 2017-01-12
Publish Date: 2017-11-15

Abstract

Abstract

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.
- Boussinesq equation,
- deformation tensor,
- eigenvalue,
- regularity estimate

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References(16)

References

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