**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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