On the global regularity for the anisotropic dissipative surface quasi-geostrophic equation

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 such as integration by parts 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.

On the boundedness of the global solution of anisotropic quasi-geostrophic equations in Sobolev space

“…The proof is divided into three steps, namely, the H 1 -uniformly bounded, the H 2uniformly bounded and finally the H s -uniformly bounded. Before starting the proof, we need some estimations of our solution which treats in [2] and gives in the following proposition: Proposition 4.1. [2] If θ 0 , α and β satisfies the assumptions stated in Theorem 2.1 and let θ be the corresponding global solution, then, for any t > 0,…”

“…Before starting the proof, we need some estimations of our solution which treats in [2] and gives in the following proposition: Proposition 4.1. [2] If θ 0 , α and β satisfies the assumptions stated in Theorem 2.1 and let θ be the corresponding global solution, then, for any t > 0,…”

“…1) We outline the main ideas in the proof of this Theorem. Since, we have the global solution of (AQG) which indicated by the author in [2] as follows:…”

“…The first studies of the system (AQG) is in [2] by Ye, who shows that this equation admits a unique global solution θ in the space C(R + , H s (R 2 )), s ≥ 2, such that…”

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In this paper we prove that the global solution of anisotropic quasi-geostrophic equation given by [2] and [8] is bounded.

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