Narrow Results

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.