Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains 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). Here, one of the novelties is that the chemotactic sensitivity SS is not a scalar function but rather attains values in R2×2R2×2, 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. We shall establish the existence of global bounded classical solutions for arbitrarily large initial data. In contrast to the corresponding case of scalar-valued sensitivities, this system does not possess any gradient-like structure due to the appearance of such matrix-valued SS. To overcome this difficulty, we will derive a series of a priori estimates involving a new interpolation inequality.To the best of our knowledge, this is the first result on global existence and boundedness in a Keller–Segel–Stokes system with tensor-valued sensitivity, in which production of the chemical signal is involved.