The fast signal diffusion limit in Keller-Segel(-fluid) systems

This paper deals with convergence of solutions to a class of parabolic Keller-Segel systems, possibly coupled to the (Navier-)Stokes equations in the framework of the full model \begin{eqnarray*} \left\{ \begin{array}{lcl} \, \, \partial_t n_{\epsilon} + u_{\epsilon} \cdot \nabla n_{\epsilon} &=& \Delta n_{\epsilon} - \nabla \cdot \Big( n_{\epsilon} S(x, n_{\epsilon}, c_{\epsilon})\cdot\nabla c_{\epsilon}\Big) + f(x, n_{\epsilon}, c_{\epsilon}), \\[1mm] \epsilon \partial_t c_{\epsilon} + u_{\epsilon}\cdot\nabla c_{\epsilon} &=& \Delta c_{\epsilon} - c_{\epsilon} + n_{\epsilon} , \\[1mm] \,\,\partial_t u_{\epsilon} + \kappa (u_{\epsilon}\cdot\nabla) u_{\epsilon} &=& \Delta u_{\epsilon} + \nabla P_{\epsilon} + n_{\epsilon} \nabla\phi, \qquad \nabla\cdot u_{\epsilon}=0 \end{array} \right. \end{eqnarray*} to solutions of the parabolic-elliptic counterpart formally obtained on taking $\epsilon\searrow 0$. In smoothly bounded physical domains $\Omega\subset {\mathbb R}^{N}$ with $N\ge 1$, and under appropriate assumptions on the model ingredients, we shall first derive a general result which asserts certain strong and pointwise convergence properties whenever asserting that supposedly present bounds on $\nabla c_{\epsilon}$ and $u_{\epsilon}$ are bounded in $L^\lambda((0,T);L^q(\Omega))$ and in $L^\infty((0,T);L^r(\Omega))$, respectively, for some $\lambda\in (2,\infty]$, $q>N$ and $r>\max\{2,N\}$ such that $\frac{1}{\lambda}+\frac{N}{2q}<\frac{1}{2}$. To our best knowledge, this seems to be the first rigorous mathematical result on a fast signal diffusion limit in a chemotaxis-fluid system.