Global well-posedness for a modified dissipative surface quasi-geostrophic equation in the critical Sobolev space H1H1

In this paper, we consider the following modified quasi-geostrophic equationsequation(MQG)∂tθ+Λαθ+u∇→θ=0,u=Λα−1R⊥(θ) where α∈]0,1[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⁎θ⁎ in the Sobolev space H1(R2)H1(R2), Eq. (MQG) has a global and smooth solution θ   in C(R+,H1(R2))C(R+,H1(R2)).