On the global regularity for the anisotropic dissipative surface quasi-geostrophic equation. (19th November 2019)

• Record Type: Journal Article
• Authors: Ye, Zhuan
• Is Part Of: Nonlinearity. Volume 33:Number 1(2020)
• Page Start: 72
• Page End: 105
• Publication Date: 2019-11-19
• Subjects: surface quasi-geostrophic equation -- anisotropic dissipation -- global regularity; 35A05 -- 35B45 -- 35B65 -- 76D03; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab41e6
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. The 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 have potential applications.