Eventual smoothness in a three-dimensional Keller-Segel-Stokes system with subquadratic degradation

Abstract This paper deals with the Keller-Segel-Stokes system with subquadratic degradation nₓ+u n= n- (n v) +rn- n^ ; vₓ+u v= v-v+n; uₓ= u+ P+n ; u=0 in a smoothly bounded convex domain R^3, where 0, r R, 0 and (1, 2). Previous literature has asserted that for all reasonably mild initial data, an associated no-flux/no-flux/Dirichlet initial-boundary value problem possesses at least one global generalized solution whenever (1, 2), but the knowledge on the regularity properties of solutions has not yet exceeded some information on fairly basic integrability features. The present study shows that these generalized solutions become eventually smooth and bounded if (53, 2) and r \ ^{2{4 - ^{2-1}}, ^2{3- } \} with some = (, , ) 0. Our result inter alia reveals that the any type of infinite-time blow-up and persisting oscillatory behavior of solutions to 3D Keller-Segel-Stokes systems with certain subquadratic degradations will never occur for the situations in which the considered population does not spontaneously proliferate, or proliferates with an adequately small rate.