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

Abstract

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_ε + u_ε \cdot \nabla n_ε &=& Δn_ε - \nabla \cdot \Big( n_ε S(x, n_ε, c_ε)\cdot\nabla c_ε\Big) + f(x, n_ε, c_ε), \\[1mm] ε\partial_t c_ε + u_ε\cdot\nabla c_ε &=& Δc_ε - c_ε + n_ε , \\[1mm] \,\,\partial_t u_ε + κ(u_ε\cdot\nabla) u_ε &=& Δu_ε + \nabla P_ε + n_ε \nablaϕ, \qquad \nabla\cdot u_ε=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.

40 pages