Global solvability and boundedness for a tumour growth chemotaxis(-fluid) model

This paper is concerned with an initial-boundary value problem for a Navier-Stokes system with a logistic source, which is posed in a two-dimensional smoothly bounded domain. Specifically, the model under consideration reads as follows:which describes the intricate interactions within a multi-component reaction-diffusion system embedded in an incompressible viscous medium, where the reaction rates satisfy μ>0, d>0, a>0, the coefficient k∈R, and the gravitational potential ϕ∈W2,∞(Ω). The paper establishes that for any k∈R, if 0<a≤1 and r>1, then the system possesses globally bounded solutions. Furthermore, we discuss the global boundedness of the model under small initial data when μ=0 and in the absence of the fluid term. It is proved that for any a>0 and d>0, there exists a small ε>0 such that for any initial data (n0,v0,w0,z0) satisfying ‖n0‖L1(Ω)+‖∇v0‖L2(Ω)+‖w0‖L1(Ω)+‖∇z0‖L2(Ω)≤ε, the solution to this problem is global and bounded in time.