Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation

We consider the following quasilinear attraction-repulsion chemotaxis system with rotation {ut=Δum−∇⋅(uS1(u,v,w,x)∇v)+∇⋅(uS2(u,v,w,x)∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,wt=Δw+γu−δw,x∈Ω,t>0,(∇um−uS1∇v+uS2∇w)⋅ν=∇v⋅ν=∇w⋅ν=0,x∈∂Ω,t>0, where Ω⊂R2 is a bounded domain with smooth boundary and α, β, γ, δ are positive constants. Here S1=(s̃ij)2×2 and S2=(sˆij)2×2 are chemosensitivity tensors with s̃ij,sˆij∈C2([0,∞)3×Ω̄), which are assumed to satisfy |S1|≤CS1 and |S2|≤CS2 with some positive constants CS1, CS2 for all (u,v,w,x)∈[0,∞)3×Ω̄. It is shown that whenever m>1, for any sufficiently smooth non-negative initial data, the system possesses at least one global bounded weak solution.