On the global regularity for anisotropic dissipative surface quasi-geostrophic equation
arXiv:1709.03073
Abstract
In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. Global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply, to overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic type Gronwall inequality, which may be of independent interest and potential applications.
To appear in Nonlinearity. We have added some more details. The new title is "On the global regularity for anisotropic dissipative surface quasi-geostrophic equation"