A 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 $θ_t + (Hθ) θ_x +(-Δ)^αθ=0$ 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.

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