Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary
Yulan Wang, Michael Winkler, Zhaoyin Xiang
Abstract
The chemotaxis-Stokes system{nt+u·∇n=Δn−∇·(n∇c),ct+u·∇c=Δc−nc,ut=Δu+∇P+n∇ϕ, ∇·u=0, (⋆)is considered in a bounded domain Ω⊂R3 with smooth boundary. The corresponding solution theory is quite well-developed in the case when (⋆) is accompanied by homogeneous boundary conditions of no-flux type for n and c, and of Dirichlet type for u. In such situations, namely, a quasi-Lyapunov structure provides regularity features sufficient to facilitate not only a basic existence theory, but also a comprehensive qualitative analysis. However, if in line with what is suggested by the modeling literature the boundary condition for the signal is changed so as to becomec(x,t)=c⋆, x∈∂Ω, t>0,with some constant c⋆≥0, then such structures apparently cease to be present at spatially global levels. The present work reveals that such properties persist at least in a weakened form of suitably localized variants, and on the basis of accordingly obtained a priori estimates it is shown that for widely arbitrary initial data an associated initial-boundary value problem for (⋆) admits a globally defined generalized solution.