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

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