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We consider the three-dimensional parabolic-parabolic Patlak-Keller-Segel equations (PKS) subject to ambient flows. Without the ambient fluid flow, the equation is super-critical in three-dimension and has finite-time blow-up solutions with arbitrarily small $L^1$-mass. In this study, we show that a family of time-dependent alternating shear flows, inspired by the clever ideas of Tarek Elgindi [39], can suppress the chemotactic blow-up in these systems.
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Time-dependent shear flows and their applications in parabolic–parabolic Patlak–Keller–Segel systems *
Abstract In this study, we investigate the behavior of three-dimensional parabolic–parabolic Patlak–Keller–Segel systems in the presence of ambient shear flows. Our findings demonstrate that when the total mass of the cell density is below a specific threshold, the solution remains globally regular as long as the flow is sufficiently strong. The primary difficulty in our analysis stems from the fast creation of chemical gradients due to strong shear advection.
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- Award ID(s):
- 2406293
- PAR ID:
- 10575659
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 38
- Issue:
- 3
- ISSN:
- 0951-7715
- Format(s):
- Medium: X Size: Article No. 035029
- Size(s):
- Article No. 035029
- Sponsoring Org:
- National Science Foundation
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