论文标题
具有各向异性分数耗散的SQG方程的全球规律性和时间衰减
Global regularity and time decay for the SQG equation with anisotropic fractional dissipation
论文作者
论文摘要
在本文中，我们专注于具有分数水平耗散和分数垂直热扩散的二维表面准地神经方程。一方面，当耗散能力仅限于合适的范围时，表面准整形方程的整体规律性将通过某些各向异性嵌入和插值不平等获得，涉及分数衍生物。一方面，我们通过各向异性插值不平等获得了全局弱解决方案的最佳较大时间衰变估计。此外，基于建立解决方案的全局$ \ dot {h}^1 $ norm所采用的参数，我们获得了以上获得的全局平滑解决方案的最佳较大时间衰减估计。最后，还得出了完整溶液和对相应线性部分的解之间差的衰减估计值。
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. On the one hand, when the dissipation powers are restricted to a suitable range, the global regularity of the surface quasi-geostrophic equation is obtained by some anisotropic embedding and interpolation inequalities involving fractional derivatives. One the one hand, we obtain the optimal large time decay estimates for global weak solutions by an anisotropic interpolation inequality. Moreover, based on the argument adopted in establishing the global $\dot{H}^1$-norm of the solution, we obtain the optimal large time decay estimates for the above obtained global smooth solutions. Finally, the decay estimates for the difference between the full solution and the solution to the corresponding linear part are also derived.
