Global existence and asymptotic stability of smooth solutions to a fluid dynamics model of biofilms in one space dimension

TL;DR

This paper proves the global existence, uniqueness, and exponential stability of smooth solutions for a one-dimensional fluid dynamics model of biofilms, demonstrating the system's dissipative nature near equilibrium.

Contribution

It provides the first analytical proof of global smooth solutions and their stability for this biofilm fluid model in one dimension.

Findings

System is hyperbolic and dissipative near equilibrium

Existence and uniqueness of global smooth solutions for small perturbations

Solutions converge exponentially to equilibrium

Abstract

In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [4] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in a open neighborhood of the physical parameters, the system is totally dissipative near its unique non vanishing equilibrium point. Using this property, we are able to prove existence and uniqueness of global smooth solutions to the Cauchy problem on the whole line for small perturbations of this equilibrium point and the solutions are shown to converge exponentially in time at the equilibrium state.