Global solutions of aggregation equations and other flows with random diffusion
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440_2022_Article_1171.pdf
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Author(s)
Rosenzweig, Matthew
Staffilani, Gigliola
Date Issued
October 31, 2022
Publisher
Springer Berlin Heidelberg
Citation
Rosenzweig, Matthew and Staffilani, Gigliola. 2022. "Global solutions of aggregation equations and other flows with random diffusion."
Version
Final published version
Abstract
Abstract
Aggregation equations, such as the parabolic-elliptic Patlak–Keller–Segel model, are known to have an optimal threshold for global existence versus finite-time blow-up. 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 Gevrey-type Fourier–Lebesgue spaces with quantifiable high probability.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00440-022-01171-8
