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arXiv · 1805.09085

A Keller-Segel-fluid system with singular sensitivity: Generalized solutions

Tobias Black·Johannes Lankeit·Masaaki Mizukami
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Abstract

In bounded smooth domains $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, we consider the Keller-Segel-Stokes system \begin{align*} n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c),\\ c_t + u\cdot \nabla c &= Δc - c + n,\\ u_t &= Δu + \nabla P + n\nabla ϕ, \qquad \nabla \cdot u=0, \end{align*} and prove global existence of generalized solutions if \[ χ<\begin{cases} \infty,&N=2,\\ \frac{5}{3},&N=3. \end{cases} \] These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.

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Tobias Black, Johannes Lankeit, Masaaki Mizukami. 2018-05-23. A Keller-Segel-fluid system with singular sensitivity: Generalized solutions. https://doi.org/10.1002/mma.5561

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