On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

Abstract

We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space Hk(wλ,κ)∩L∞, where k=max⁡(0,3/2−α) and wλ,κ is a given family of Muckenhoupt weights, we prove a global existence result in the subcritical case α∈(1,2). We also prove a local existence theorem for large data in H2(wλ,κ)∩L∞ in the supercritical case α∈(0,1). The proofs are based on the use of the weighted Littlewood–Paley theory, interpolation along with some new commutator estimates.