Global weak solutions to a chemotaxis-Navier-Stokes system in $ \mathbb{R}^3 $

Abstract

<p style='text-indent:20px;'>The Cauchy problem in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula> for the chemotaxis-Navier–Stokes system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{l} n_t + u\cdot\nabla n = \Delta n - \nabla \cdot (n\nabla c), \\ c_t + u\cdot\nabla c = \Delta c - nc, \\ u_t + (u\cdot\nabla) u = \Delta u + \nabla P + n\nabla\phi, \qquad \nabla \cdot u = 0, \ \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered. Under suitable conditions on the initial data <inline-formula><tex-math id="M3">\begin{document}$ (n_0, c_0, u_0) $\end{document}</tex-math></inline-formula>, with regard to the crucial first component requiring that <inline-formula><tex-math id="M4">\begin{document}$ n_0\in L^1( \mathbb{R}^3) $\end{document}</tex-math></inline-formula> be nonnegative and such that <inline-formula><tex-math id="M5">\begin{document}$ (n_0+1)\ln (n_0+1) \in L^1( \mathbb{R}^3) $\end{document}</tex-math></inline-formula>, a globally defined weak solution with <inline-formula><tex-math id="M6">\begin{document}$ (n, c, u)|_{t = 0} = (n_0, c_0, u_0) $\end{document}</tex-math></inline-formula> is constructed. Apart from that, assuming that moreover <inline-formula><tex-math id="M7">\begin{document}$ \int_{ \mathbb{R}^3} n_0(x) \ln (1+|x|^2) dx $\end{document}</tex-math></inline-formula> is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.</p>