# On the loss of continuity for supercritical drift-diffusion equations

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We consider the (linear) drift-diffusion equation $\partial_t \theta + u \cdot \nabla \theta + (-\Delta)^s \theta = 0$. Here the divergence free drift $u$ belongs to a supercritical space, and $0 < s \leq 1$. We prove that starting with smooth initial data solutions may become discontinuous in ﬁnite time. For $s < 1$ this may even be achieved with autonomous drift. On the other hand, for $s = 1$ and autonomous drift, in two space dimensions we obtain a modulus of continuity for the solution depending only on the $L^1$ norm of the drift, which is a supercritical quantity. This is joint work with L. Silvestre and A. Zlatos.