CRM: Centro De Giorgi
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ERC Workshop on Geometric Partial Differential Equations

On the continuity (or discontinuity) of solutions of drift-diffusion equations

speaker: Luis Silvestre (The University of Chicago)

abstract: We study parabolic equations which consist on a drift term (a vector field times the gradient) plus a diffusion term (either the laplacian or the fractional Laplacian). We analyze what assumptions on the drift would assure that the solution remains continuous for positive time. A particularly interesting case is when the drift is a divergence free vector field, since these appear frequently in equations from fluid mechanics. Assuming that the drift is bounded respect to some norm which is invariant by the scaling of the equation gives Holder continuity estimates in many cases. We will prove that when this scaling condition is violated, discontinuities can form in finite time, even if the drift is divergence free. A notable exception is for an equation with classical diffusion (with the usual Laplacian), in 2 space dimensions, and a drift which is independent of time. For that case a modulus of continuity is obtained for any divergence free drift in $L1$ (which is highly supercritical).

Tue 11 Sep, 9:30 - 10:15, Aula Dini
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