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Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux

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Abstract

In this paper, we investigate the 3D Keller–Segel–Stokes (K–S–S) system with nonlinear diffusion term \(\Delta n^{m}\) (\(m>0\)) and rotational flux posed in a bounded domain \(\Omega \) with smooth boundary. Under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity S(x, n, c) satisfies \(|S(x,n,c)|\le C_{S}(1+n)^{-\alpha }\), by seeking some new functionals and using the bootstrap arguments on the regularized system, we establish the existence and boundedness of global weak solutions to K–S–S system for arbitrarily large initial data under the assumption \(m+2\alpha >2\) and \(m>\frac{3}{4}\), which includes both the degenerate \((m>1)\) and the singular \((m<1)\) case.

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  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Yingping Peng & Zhaoyin Xiang

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  1. Yingping Peng
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  2. Zhaoyin Xiang
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Peng, Y., Xiang, Z. Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux. Z. Angew. Math. Phys. 68, 68 (2017). https://doi.org/10.1007/s00033-017-0816-6

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  • DOI: https://doi.org/10.1007/s00033-017-0816-6

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