Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: The 3D case

In this paper we continue to deal with the initial–boundary value problem for the coupled Keller–Segel–Stokes system{nt+u⋅∇n=Δn−∇⋅(nS(x,n,c)⋅∇c),(x,t)∈Ω×(0,T),ct+u⋅∇c=Δc−c+n,(x,t)∈Ω×(0,T),ut+∇P=Δu+n∇ϕ,(x,t)∈Ω×(0,T),∇⋅u=0,(x,t)∈Ω×(0,T), where Ω⊂RdΩ⊂Rd is a bounded domain with smooth boundary and the chemotactic sensitivity SS is not a scalar function but rather attains values in Rd×dRd×d, and satisfies |S(x,n,c)|≤CS(1+n)−α|S(x,n,c)|≤CS(1+n)−α with some CS>0CS>0 and α>0α>0. When d=2d=2, our previous work (J. Differential Equations, 2015) has established the existence of global bounded classical solutions under the subcritical assumption α>0α>0, which is consistent with the corresponding results of the fluid-free system, but the method seems to be invalid in the three-dimensional setting.In this paper, for the case d=3d=3, we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption α>12, which is slightly stronger than the corresponding subcritical assumption α>13 on the fluid-free system, where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions. The key idea here is to establish the general LpLp regularity of u   from a rather low LpLp regularity of n, which will be obtained by a new combinational functional.