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The fast signal diffusion limit in Keller-Segel(-fluid) systems. (English) Zbl 1426.92009

Summary: 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 \[\begin{cases}\partial _t n_\varepsilon + u_\varepsilon \cdot \nabla n_\varepsilon = \Delta n_\varepsilon - \nabla \cdot \Big ( n_\varepsilon S(x,n_\varepsilon ,c_\varepsilon )\cdot \nabla c_\varepsilon \Big ) + f(x,n_\varepsilon ,c_\varepsilon ), \\ \varepsilon \partial _t c_\varepsilon + u_\varepsilon \cdot \nabla c_\varepsilon = \Delta c_\varepsilon - c_\varepsilon + n_\varepsilon , \\ \partial _t u_\varepsilon + \kappa (u_\varepsilon \cdot \nabla )u_\varepsilon = \Delta u_\varepsilon + \nabla P_\varepsilon + n_\varepsilon \nabla \phi , \qquad \nabla \cdot u_\varepsilon =0, \end{cases}\] 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 \(\varepsilon >0\).

MSC:

92C17 Cell movement (chemotaxis, etc.)
35Q30 Navier-Stokes equations
35K55 Nonlinear parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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