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arXiv · 2404.15674

Enhanced dissipation and blow-up suppression for an aggregation equation with fractional diffusion and shear flow

Niu Binqian·Binbin Shi·Weike Wang
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Abstract

In this paper, we consider an aggregation equation with fractional diffusion and large shear flow, which arise from modelling chemotaxis in bacteria. Without the advection, the solution of aggregation equation may blow up in finite time. First, we study the enhanced dissipation of shear flow by resolvent estimate method, where the fractional Laplacian $(-\Delta)^{\alpha/2}$ is considered and $\alpha\in (0,2)$. Next, we show that the enhanced dissipation of shear flow can suppress blow-up of solution to aggregation equation with fractional diffusion and establish global classical solution in the case of $\alpha\geq 3/2$. Here we develop some new technical to overcome the difficult of low regularity for fractional Laplacian.

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Niu Binqian, Binbin Shi, Weike Wang. 2024-04-24. Enhanced dissipation and blow-up suppression for an aggregation equation with fractional diffusion and shear flow. https://arxiv.org/abs/2404.15674

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