The global solvability of the Cauchy problem for a multi-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization

Abstract

This paper is dedicated to the analysis of the Cauchy problem for a chemotaxis-Navier–Stokes system modeling coral fertilization in spatial dimensions two and three. We first present the unique local solvability of a smooth solution to the system for large initial data and then establish some blow-up criteria of the solution. In the whole plane, the global existence of smooth solutions to the model for a large class of initial data is constructed. Finally, we also prove the global existence of smooth solutions under the some explicit smallness conditions of initial data. In particular, we present the time decay rates of the solution in L∞ by using the De Giorgi method.