On the loss of continuity for super-critical drift-diffusion equations

Abstract

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts, that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local $L^1$ norm of the drift, which is a super-critical quantity.

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References in corpus (3)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Global well-posedness of slightly supercritical active scalar equations
• Exit times of diffusions with incompressible drift

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