Global well-posedness for the critical dissipative quasi-geostrophic equations in L∞

In this paper, we study the critical dissipative quasi-geostrophic equations in scaling invariant spaces. We prove that there exists a global-in-time small solution for small initial data View the MathML sourceθ0∈L∞∩Ḣ1 such that ℛ(θ0)∈L∞ℛ(θ0)∈L∞, where ℛℛ is the Riesz transform. As a corollary, we prove that if in addition, View the MathML sourceθ0∈Ḃ∞,q0, 1≤q<21≤q<2, is small enough, then View the MathML sourceθ∈L̃t∞Ḃ∞,q0∩L̃t1Ḃ∞,q1.