Eventual smoothness and asymptotic stabilization in a two-dimensional logarithmic chemotaxis-Navier–Stokes system with nutrient-supported proliferation and signal consumption

In this paper, we study the influence of nutrient-dependent cell proliferation on large time behavior of the solutions to an associated initial-boundary problem of [Formula presented] subject to no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain Ω⊂R², where +nc in the first equation denotes cell reproduction in dependence on nutrients. It is proved that the initial-boundary problem is globally solvable in a generalized sense for any appropriately regular initial data, and that the generalized solutions thereof emanating from suitably small initial data will become smooth from a finite waiting time and exponentially approach [Formula presented] as t→∞. As compared to a precedent result obtained without cell proliferation, the complexity caused by +nc in the first equation requires some conditions on the initial data c₀ rather only on n₀ in order to derive the large time behavior of the solutions thereof, which seems comprehensible because of the necessity for offsetting the possibly increasing trend of cell by +nc.