Boundedness in a two-dimensional doubly degenerate nutrient taxis system

In this work, we study the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u v \nabla u)-χ\nabla \cdot\left(u^{2} v \nabla v\right)+\ell u v, & x \in Ω, t>0, \ v_t=Δv-u v, & x \in Ω, t>0 \end{cases} \end{align} in a smoothly bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$, where $\chi >0$ and $\mathrm{\ell }\ge 0$. In this paper, we present that for all reasonably regular initial data, the model \eqref{eq 0.1} possesses a global bounded weak solution which is continuous in its first and essentially smooth in its second component. \end{abstract}