Global well-posedness of the transport equation with nonlocal velocity in Besov spaces with critical and supercritical dissipation

We study the transport equation with nonlocal velocity introduced in Córdoba et al (2005 Ann. Math. 162 1377-89). We prove its global well-posedness under critical and supercritical dissipation, the last case under the smallness condition, in Besov spaces with critical and subcritical regularity indexes, using the Fourier localization method and modulus of continuity.