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Global existence and boundedness of weak solutions to a chemotaxis–Stokes system with rotational flux term

  • Feng LiEmail author
  • Yuxiang Li
Article
  • 23 Downloads

Abstract

In this paper, the three-dimensional chemotaxis–Stokes system
$$\begin{aligned} \left\{ \begin{array}{ll} n_{t}+u\cdot \nabla n=\Delta n^m-\nabla \cdot (n S(x,n,c)\cdot \nabla c),&{}\quad x\in \Omega ,\ \ t>0,\\ c_t+u\cdot \nabla c=\Delta c-nf(c),&{}\quad x\in \Omega ,\ \ t>0,\\ u_t+\nabla P=\Delta u +n\nabla \phi ,&{}\quad x\in \Omega ,\ \ t>0,\\ \nabla \cdot u=0, &{}\quad x\in \Omega ,\ \ t>0, \end{array}\right. \end{aligned}$$
posed in a bounded domain \(\Omega \subset {\mathbb {R}}^3\) with smooth boundary is considered under the no-flux boundary condition for n, c and the Dirichlect boundary condition for u under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity S(xnc) satisfies \(S(x,n,c)<n^{l-2}\widetilde{S}(c)\) with \(l>2\) for some non-decreasing function \(\widetilde{S}\in C^{2}([0,\infty ))\). In the present work, it is shown that the weak solution is global in time and bounded while \(m>m^\star (l)\), where
$$\begin{aligned} m^\star (l)= \left\{ \begin{array}{ll} l-\frac{5}{6},&{}\quad \mathrm {if}\ \ \frac{31}{12}\ge l>2,\\ \frac{7}{5}l-\frac{28}{15}, &{}\quad \mathrm {if}\ \ l>\frac{31}{12}. \end{array}\right. \end{aligned}$$

Keywords

Chemotaxis Porous media diffusion Tensor-valued sensitivity Global existence 

Mathematics Subject Classification

35K92 35Q35 35Q92 92C17 

Notes

Acknowledgements

The authors convey sincere gratitude to the anonymous referees for their careful reading of this manuscript and valuable comments which greatly improve the exposition of the paper.

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Authors and Affiliations

  1. 1.School of MathematicsSoutheast UniversityNanjingPeople’s Republic of China

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