Narrow Results

We consider the coupled chemotaxis–Navier–Stokes system with logistic source term

In this work, we consider the two-species chemotaxis system with Lotka–Volterra competitive kinetics in a bounded domain with smooth boundary. We construct weak solutions and prove that they become smooth after some waiting time. In addition, the asymptotic behavior of the solutions is studied. Our results generalize some well-known results in the literature.

This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well.

This paper reconsiders the Keller–Segel–Navier–Stokes model with indirect signal production and weak logistic-type degradation in a three-dimensional (3D) bounded and smooth domain. One recent literature has asserted that for all reasonably regular initial data, the associated initial-boundary value problem with certain sub-quadratic degradation possesses at least one global generalized solution which is uniformly bounded and converges to the constant equilibrium in $L^1(\mathrm{\Omega })\times L^p(\mathrm{\Omega })\times L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega };{ℝ}^3)$ with any $p\in [1,\infty )$. Nevertheless, the knowledge on regularity properties of solution has not yet exceeded some information on fairly basic integrability features. The present study firstly elevates the regularities of some solution components under the same assumption on degradation exponent, especially establishes the global boundedness in $L^1(\mathrm{\Omega })\times L^{\infty }(\mathrm{\Omega })\times L^{max\{2,\frac{3\alpha }{5-2\alpha }\}}(\mathrm{\Omega })\times L^2(\mathrm{\Omega };{ℝ}^3)$ with $\alpha \in (\frac{4}{3},2)$, and secondly asserts that each of these generalized solutions becomes eventually classical and bounded in $L^{\infty }(\mathrm{\Omega })\times W^{1,\infty }(\mathrm{\Omega })\times W^{1,\infty }(\mathrm{\Omega })\times L^{\infty }(\mathrm{\Omega };{ℝ}^3)$ under some suitably strong sub-quadratic degradation assumption and an explicit smallness condition on the intrinsic growth rate of cells relative to some powers of the degradation coefficient. As a by-product of the latter, these solutions are shown to stabilize toward the corresponding spatially homogeneous state in $L^p(\mathrm{\Omega })\times W^{1,p}(\mathrm{\Omega })\times W^{1,p}(\mathrm{\Omega })\times L^p(\mathrm{\Omega };{ℝ}^3)$. Our results indeed provide a more in-depth understanding on the global dynamics of solutions, and significantly improve previously known ones. In comparison to the related contributions in the case of direct signal production, our findings inter alia rigorously reveal that the indirect signal production mechanism results in some genuine regularizing effects in the sense that the possibly destabilizing action of chemotactic cross-diffusion in the 3D Keller–Segel–Navier–Stokes system can be greatly weakened or even completely excluded by this indirectness.