Aggregation equations, such as the parabolicelliptic Patlak–Keller–Segel model, are known to have an optimal threshold for global existence versus finitetime blowup. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Math Res Not IMRN 23:9370–9385, 2020) showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevreytype Fourier–Lebesgue spaces with quantifiable high probability.
 Award ID(s):
 1927258
 NSFPAR ID:
 10339534
 Date Published:
 Journal Name:
 IMA Journal of Applied Mathematics
 Volume:
 86
 Issue:
 6
 ISSN:
 02724960
 Page Range / eLocation ID:
 1349 to 1396
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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