LAUSR

This paper deals with a two‐competition‐species chemotaxis‐Navier‐Stokes system with two different consumed signals (n1)t+u·∇n1=d1Δn1−χ1∇·(n1∇c)+μ1n1(1−n1−a1n2),inΩ×(0,∞),ct+u·∇c=d2Δc−α1cn2,inΩ×(0,∞),(n2)t+u·∇n2=d3Δn2−χ2∇·(n2∇v)+μ2n2(1−a2n1−n2),inΩ×(0,∞),vt+u·∇v=d4Δv−α2vn1,inΩ×(0,∞),ut+(u·∇)u=Δu+∇P+(β1n1+β2n2)∇ϕ,inΩ×(0,∞),∇·u=0,inΩ×(0,∞), in a smooth bounded domain Ω⊂R3 under zero Neumann boundary conditions for n1,n2,c,v , and homogeneous Dirichlet boundary condition for u , where the parameters di ( i=1,2,3,4 ) and χj,μj,aj,αj,βj ( j=1,2 ) are positive. This system describes the evolution of two‐competing species which react on two different chemical signals in a liquid surrounding environment. Recently, the boundedness and stabilization of classical solutions to the above system under two‐dimensional case have been derived in the previous works. However, to the best of our knowledge, the well‐posedness problem of solutions for the above system is still open in the three dimensional setting, because of the difficulties in the Navier‐Stokes system. The aim of this paper is to construct global weak solutions and show that after some waiting time, these weak solutions become eventually smooth.