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Uniqueness of Weak Solutions to One-Dimensional Doubly Degenerate Cross-Diffusion System

Xiuqing Chen, Bang Du

TL;DR

This work proves the uniqueness of global weak solutions for a one-dimensional doubly degenerate cross-diffusion system modeling feeding bacteria in nutrient-poor environments, under the condition $1/u^0\in L^1(\Omega)$. The authors introduce the antiderivative $w(t,x)=\int_0^x (u+v)\,dy$, which converts the problem into a coupled $(w,v)$ framework and enables energy-type estimates. The main contribution is a two-stage estimate that bounds $\|w_1-w_2\|_{L^2}$ and $\|v_1-v_2\|_{L^2}$, leveraging mollification, boundary cancellations, and a Gronwall argument to deduce uniqueness. This advances the mathematical understanding of cross-diffusion systems in low dimensions and provides insight into the bacterial dynamics in malnourished environments.

Abstract

The uniqueness of global weak solutions to one-dimensional doubly degenerate cross-diffusion system is shown. The equations model the evolution of feeding bacterial populations in a malnourished environment. The key idea of the proof is applying anti-derivative of the sum of weak solutions to the system.

Uniqueness of Weak Solutions to One-Dimensional Doubly Degenerate Cross-Diffusion System

TL;DR

This work proves the uniqueness of global weak solutions for a one-dimensional doubly degenerate cross-diffusion system modeling feeding bacteria in nutrient-poor environments, under the condition . The authors introduce the antiderivative , which converts the problem into a coupled framework and enables energy-type estimates. The main contribution is a two-stage estimate that bounds and , leveraging mollification, boundary cancellations, and a Gronwall argument to deduce uniqueness. This advances the mathematical understanding of cross-diffusion systems in low dimensions and provides insight into the bacterial dynamics in malnourished environments.

Abstract

The uniqueness of global weak solutions to one-dimensional doubly degenerate cross-diffusion system is shown. The equations model the evolution of feeding bacterial populations in a malnourished environment. The key idea of the proof is applying anti-derivative of the sum of weak solutions to the system.
Paper Structure (2 sections, 5 theorems, 43 equations)

This paper contains 2 sections, 5 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm_main']}

Key Result

Theorem 1

Let $(u,v)$ be a weak solution to eqt-reg_init_time satisfying eqt_u_weak-est. If $1/u^0 \in L^1(\Omega)$, then $(u,v)$ is unique.

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof