Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars
| | Speaker: | Matthew Rosenzweig， MIT
| | Inviter: | | Title:
| Global solutions of aggregation equations and other flows with random diffusion | Time & Venue:
| 2021.10.28 21:00-21:50 Zoom Meeting ID: 982 4274 5864 Passcode: 1m3Lkm | Abstract:
| Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. 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. In this talk, we investigate whether 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.
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