Boundedness in a two-dimensional doubly degenerate nutrient taxis system

TL;DR

This paper proves the existence of global bounded weak solutions for a two-dimensional doubly degenerate nutrient taxis system, demonstrating regularity and boundedness under certain initial conditions.

Contribution

It establishes the first proof of global bounded weak solutions for this complex nutrient taxis model in two dimensions.

Findings

Existence of global bounded weak solutions

Solutions are continuous in the first component and smooth in the second

Model behavior is well-posed for regular initial data

Abstract

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)-\chi \nabla \cdot\left(u^{2} v \nabla v\right)+\ell u v, & x \in \Omega, t>0, \\ v_t=\Delta v-u v, & x \in \Omega, t>0 \end{cases} \end{align} in a smoothly bounded convex domain $\Omega \subset \mathbb{R}^2$, where $\chi>0$ and $\ell \geq 0$. 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}