Abstract
In this paper, we study Cauchy problem of the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Taking advantage of a coupling structure of the equations and using the damping effect of the growth term \begin{document}$ g(n) $\end{document} , we obtain the necessary estimates of solution \begin{document}$ (n,c,u) $\end{document} without the diffusion term \begin{document}$ \Delta n $\end{document} . These uniform estimates enable us to establish the global-in-time existence of almost weak solutions for the system.
Highlights
The Cauchy problem of the two-dimensional incompressible chemotaxis-Navier-Stokes system with partial diffusion reads ∂tn + u · ∇n = −∇ (n∇c) g(n), ∂tc ∇c
When κ = 0, a global existence result of weak solutions obtained by Duan, Lorz and Markowich in [3] for problem (1.2) under smallness assumptions on either ∇Φ or the initial data c0
Compared with (1.2), we obtain the global existence of weak solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion
Summary
When κ = 0, a global existence result of weak solutions obtained by Duan, Lorz and Markowich in [3] for problem (1.2) under smallness assumptions on either ∇Φ or the initial data c0. Zhang and Zheng [35] established some new estimates and proved the global well-posedness of energy solution for the two-dimensional chemotaxis-Navier-Stokes equations in R2 for the rough initial data. Compared with (1.2), we obtain the global existence of weak solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. After that by establishing a priori estimates and using the Arzela-Ascoli theorem, we prove the global existence of weak solutions for the system we studied.
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