Abstract
The chemotaxis-Navier—Stokes system
is considered in a smoothly bounded planar domain Ω under the boundary conditions
with a given nonnegative constant C✭. It is shown that if (n0, c0, u0) is sufficiently regular and such that the product \({\left\| {{n_0}} \right\|_{{L^1}\left(\Omega \right)}}\left\| {{c_0}} \right\|_{{L^\infty}\left(\Omega \right)}^2\) is suitably small, an associated initial value problem possesses a bounded classical solution with (n, c, u)|t=0 = (n0, c0, u0).
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The authors are very grateful to the referees for their helpful suggestions.
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Y. Wang was supported by the Sichuan Science and Technology program (Grant No. 2021ZYD0008) and Sichuan Youth Science and Technology Foundation (Grant No. 2021JDTD0024). M. Winkler acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (Project No. 411007140, GZ: WI 3707/5-1). Z. Xiang was supported by the NNSF of China (Grant Nos. 11971093, 11771045), the Applied Fundamental Research Program of Sichuan Province (Grant No. 2020YJ0264), and the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2019J096)
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Wang, Y.L., Winkler, M. & Xiang, Z.Y. A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier—Stokes System involving Dirichlet Boundary Conditions for the Signal. Acta. Math. Sin.-English Ser. 38, 985–1001 (2022). https://doi.org/10.1007/s10114-022-1093-7
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DOI: https://doi.org/10.1007/s10114-022-1093-7