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 $H^k(w_{\lambda ,\kappa })\cap L^{\infty },$ with $k=\max (0,3/2-\alpha )$ and $w_{\lambda ,\kappa }$ is a given family of Muckenhoupt weights. We prove a global existence result in the subcritical case $\alpha \in (1,2)$. We also prove a local existence theorem for large data in $H^2(w_{\lambda ,\kappa })\cap L^{\infty }$ in the supercritical case $\alpha \in (0,1)$. The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.

arXiv comment18 pages