Wellposedness of solution for an N-D chemotaxis-convection model during tumor angiogenesis

In this paper, we consider the following parabolic-parabolic-elliptic system } {align*} \{ & u_t=Δ u-∇·(u∇ v)+ξ∇·(u∇ w)+au-μ uα, && x∈Ω, t>0,\\ & v_t=Δ v+∇·(v∇ w)-v+u,&& x∈Ω, t>0,\\ & 0=Δ w-w+u,&& x∈Ω, t>0\\ . {align*} on a bounded domain Ω⊂ RN (N≥1) with smooth boundary ∂ Ω, where μ, a, α are positive constants and ξ. If one of the following cases holds:\\ (i) N≥4 and α>4N-4+N√2N²-6N+82N;\\ (ii) $N=3$, α>2, for any μ>0 or α=2, the index μ should be suitably big;\\ (iii) $N=2$, α≥2, for any μ>0.\\ Without any restriction on the index ξ, for any given suitably regular initial data, the corresponding Neumann initial-boundary problem admits a unique global and bounded classical solution.