L∞ bounds in a two-dimensional doubly degenerate nutrient taxis system with general cross-diffusive flux

The doubly degenerate nutrient taxis model{ut=∇⋅(uv∇u)−∇⋅(S(u)v∇v)+ℓuv,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, is considered under no-flux boundary conditions in a bounded convex planar domain Ω with smooth boundary. It is firstly shown that within a general framework involving a conveniently large class of nonlinearities S, under an a priori assumption on regularity of solutions to certain approximate problems one can construct a global weak solution whose first component is uniformly bounded. Applications of this to particular settings thereafter reveal global existence of such bounded weak solutions when either S∈C1([0,∞)) satisfieslimsupξ→∞S(ξ)ξα<∞with some α<2 and the initial data (u0,v0) are reasonably regular but arbitrary large, or S(ξ)=ξ2 for ξ≥0 and the initial data are such that v0 satisfies an appropriate smallness condition.