GLOBAL SOLUTIONS TO CHEMOTAXIS-NAVIER-STOKES EQUATIONS IN CRITICAL BESOV SPACES.

In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential ϕ and the small initial data (u0,n0,c0) in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants σ 0 and C0 such that if the gravitational potential ϕ ∈B_p³,/₁^p (R³) and the initial data (u0,n0,c0):=(u₀¹,u₀²,u₀³,n0,c0):=(u₀^h,u₀³,n0,c0) satisfies . . . for some p; q with …, then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field u₀³ in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system. [ABSTRACT FROM AUTHOR]