On the Loss of Continuity for Super-Critical Drift-Diffusion Equations

By Luis Silvestre, Vlad Vicol, Andrej Zlatoš. 2012

75 citations recorded by the source corpus.

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 L1 norm of the drift, which is a super-critical quantity.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d9b0e371eab30ea49f7129691e4b07f61ec112fd

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Critical SQG in Bounded Domains (2016, background): “replaced by Λs with 0 <s≤ 1 then if θis bounded in Cα with α>1 − s, then the solution is smooth ([12]). (This condition is sharp, if one considers general linear advection diffusion equations, ([23]). In [11] the smallness of αis used to show that the term corresponding to Qin the ﬁnite differ”