Communications in Mathematical Sciences

Volume 15 (2017)

Number 2

Discrete model for nonlocal transport equations with fractional dissipation

Pages: 289 – 303

DOI: http://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 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

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