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

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _t n_\varepsilon + u_\varepsilon \cdot \nabla n_\varepsilon &{}=&{} \Delta n_\varepsilon - \nabla \cdot \Big ( n_\varepsilon S(x,n_\varepsilon ,c_\varepsilon )\cdot \nabla c_\varepsilon \Big ) + f(x,n_\varepsilon ,c_\varepsilon ), \\ \varepsilon \partial _t c_\varepsilon + u_\varepsilon \cdot \nabla c_\varepsilon &{}=&{} \Delta c_\varepsilon - c_\varepsilon + n_\varepsilon , \\ \partial _t u_\varepsilon + \kappa (u_\varepsilon \cdot \nabla )u_\varepsilon &{}=&{} \Delta u_\varepsilon + \nabla P_\varepsilon + n_\varepsilon \nabla \phi , \qquad \nabla \cdot u_\varepsilon =0 \end{array}\right. \end{aligned}$$

to solutions of the parabolic–elliptic counterpart formally obtained on taking $\varepsilon {\searrow } 0$. 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 $\nabla c_\varepsilon$ and $u_\varepsilon$ 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. This general result will thereafter be concretized in the context of two examples: firstly, for an unforced Keller–Segel–Navier–Stokes system we shall establish a statement on global classical solutions under suitable smallness conditions on the initial data, and show that these solutions approach a global classical solution to the respective parabolic–elliptic simplification. We shall secondly derive a corresponding convergence property for arbitrary solutions to fluid-free Keller–Segel systems with logistic source terms, which in spatially one-dimensional settings turn out to allow for a priori estimates compatible with our general theory. Building on the latter in conjunction with a known result on emergence of large densities in the associated parabolic–elliptic limit system, we will finally discover some quasi-blowup phenomenon for the fully parabolic Keller–Segel system with logistic source and suitably small parameter $\varepsilon >0$.