GENERALIZED SOLUTIONS TO A CHEMOTAXIS-NAVIER STOKES SYSTEM WITH ARBITRARY SUPERLINEAR DEGRADATION

In this work, we study a chemotaxis-Navier Stokes model in a two-dimensional setting as follows: {nt u " Vn = V " (nVe)+ f (n); et u " Ve = c +n; ut +K(u " V)u = + VP + nVO; V " u = 0}. Motivated by a recent work due to Winkler, we aim at investigating generalized solvability for the model without imposing a critical superlinear exponent restriction on the logistic source function f. Specifically, it is proven in the present work that there exists a triple of integrable functions (n, c, u) solving the system globally in a generalized sense provided that f E Cl ([O, Do)) satisfies f (0) > 0 and f (n) < rn n7 (n > 0) with any 7 > 1. Our result indicates that persistent Dirac-type singularities can be ruled out in our model under the aforementioned mild assumption on f. After giving the existence result for the system, we also show that the generalized solution exhibits eventual smoothness as long as it/r is sufficiently large.