Communications in Mathematical Sciences

Volume 15 (2017)

Number 2

Discrete model for nonlocal transport equations with fractional dissipation

Pages: 289 – 303

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n2.a1

Author

Tam Do (Rice Unversity, Houston, Texas, U.S.A.)

Abstract

In this note, we propose a discrete model to study one-dimensional transport equations with non-local drift and supercritical dissipation. The inspiration for our model is the equationθt+(Hθ)θx+(Δ)αθ=0,where H is the Hilbert transform. For our discrete model, we present blow-up results that are analogous to the known results for the above equation. In addition, we will prove regularity for our discrete model which suggests supercritical regularity in the range 1/4<α<1/2 in the continuous setting.

Keywords

nonlocal transport, dyadic model, supercritical regularity

2010 Mathematics Subject Classification

35Q35

Published 21 February 2017