Maximum and minimum principles for nonlinear transport equations on discrete-space domains

Abstract

We consider nonlinear scalar transport equations on the domain with discrete space and continuous time. As a motivation we derive a conservation law on these domains. In the main part of the paper we prove maximum and minimum principles that are later applied to obtain an a priori bound which is applied in the proof of existence of solution and its uniqueness. Further, we study several consequences of these principles such as boundedness of solutions, sign preservation, uniform stability and comparison theorem which deals with lower and upper solutions.

Keywords

• Nonlinear transport equation
• discrete-space domain
• maximum principles
• existence
• uniqueness
• sign preservation
• uniform stability
• nonlinear comparison theorem