Sur l'existence locale pour une équation de scalaires actifs

We address the local well-posedness for the active scalar equation ∂_tθ+u{dot operator}∇θ=0 where u=-∇^⊥(-δ)^-1+β/2θ. This equation reduces to the Euler equation if β=0 and to the quasi-geostrophic equation for β=1. In this note, we prove the local existence for the equation in the space H^1+β+ε, where ε>0, for 1<β<2. An earlier result by Chae, Constantin, Córdoba, Gancedo, and Wu shows the local existence in H⁴. The improvement is due to a sharper commutator estimate, while the fractional exponent is obtained through a different treatment of the nonlinearity using a double commutator inequality.