Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation

Abstract

<p style='text-indent:20px;'>We study the chemotaxis–Navier–Stokes system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\;\; \begin{aligned} n_t + u \cdot \nabla n &amp;\;\; = \;\; \Delta n - \nabla \cdot (nS(x,n,c) \nabla c), \\ c_t + u\cdot \nabla c &amp;\;\; = \;\; \Delta c - n f(c), \\ u_t + (u\cdot \nabla) u &amp;\;\; = \;\; \Delta u + \nabla P + n \nabla \phi, \;\;\;\;\;\; \nabla \cdot u = 0, \;\;\;\;\;\; \end{aligned} \right. \tag{$\star$} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>with no-flux boundary conditions for <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ c $\end{document}</tex-math></inline-formula> and Dirichlet boundary conditions for <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> in a bounded, convex, smooth domain <inline-formula><tex-math id="M4">\begin{document}$ \Omega \subseteq \mathbb{R}^2 $\end{document}</tex-math></inline-formula>, which is motivated by recent modeling approaches from biology for aerobic bacteria suspended in a sessile water drop. We further do not assume the chemotactic sensitivity <inline-formula><tex-math id="M5">\begin{document}$ S $\end{document}</tex-math></inline-formula> to be scalar as is common, but to be able to attain values in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^{2\times2} $\end{document}</tex-math></inline-formula>, which allows for more complex modeling of bacterial behavior. <p style='text-indent:20px;'>While there have been various results for scalar <inline-formula><tex-math id="M7">\begin{document}$ S $\end{document}</tex-math></inline-formula> and some for the non-scalar case with only a Stokes fluid equation simplifying the analysis of the third equation in (<inline-formula><tex-math id="M8">\begin{document}$ \star $\end{document}</tex-math></inline-formula>), we consider the fully combined case giving us very little to go on in terms of a priori estimates. We nonetheless manage to still achieve sufficient estimates using Trudinger–Moser type inequalities to extend the existence results seen in a recent work by Winkler for the Stokes case with non-scalar <inline-formula><tex-math id="M9">\begin{document}$ S $\end{document}</tex-math></inline-formula> to the full Navier–Stokes case. Namely, we construct a similar global mass-preserving solution for (<inline-formula><tex-math id="M10">\begin{document}$ \star $\end{document}</tex-math></inline-formula>) in planar convex domains under fairly weak assumptions on the parameter functions.