Discrete model for nonlocal transport equations with fractional dissipation

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 equation\[\theta_t + (H \theta) \theta_x + {(-\Delta)}^{\alpha} \theta = 0,\]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 $1/4 \lt \alpha \lt 1/2$ in the continuous setting.

Keywords

nonlocal transport, dyadic model, supercritical regularity

2010 Mathematics Subject Classification

35Q35

Full Text (PDF format)

Published 21 February 2017