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.
- Award ID(s):
- 2006372
- PAR ID:
- 10325823
- Editor(s):
- Berselli, Luigi G; Ruzicka, Michael
- Date Published:
- Journal Name:
- Progress in mathematical fluid dynamics, Lecture Notes in Mathematics
- Volume:
- 2272
- Page Range / eLocation ID:
- 125-161
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work [ 90] by the second and third named authors showed that with quantifiable high probability, random diffusion restores 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 the Patlak–Keller–Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from [ 90] in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow \infty $.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-54899
