Global well-posedness for a modified dissipative surface quasi-geostrophic equation in the critical Sobolev space [formula omitted

Abstract

In this paper, we consider the following modified quasi-geostrophic equations (MQG) ∂ t θ + Λ α θ + u ∇ → θ = 0 , u = Λ α − 1 R ⊥ ( θ ) where α ∈ ] 0 , 1 [ is a fixed parameter. This equation was recently introduced by P. Constantin, G. Iyer and J. Wu (2001) in [4] as a modification of the classical quasi-geostrophic equation. In this paper, we prove that for any initial data θ * in the Sobolev space H 1 ( R 2 ) , Eq. (MQG) has a global and smooth solution θ in C ( R + , H 1 ( R 2 ) ) .