NON-EXISTENCE OF A GLOBALLY CONVEX ENTROPY PAIR AND WAVE STRUCTURE FOR A FLUID DYNAMICS MODEL OF BIOFILMS

Abstract

We study the hyperbolic structure of the one-dimensional fluid dynamics system introduced by Clarelli, Di Russo, Natalini, and Ribot (2013) to model biofilm growth, and subsequently analyzed analytically by Bianchini and Natalini (2016), who proved global existence and exponential stability of smooth solutions near the unique equilibrium by exploiting a total dissipativity condition in lieu of a convex entropy. The absence of a globally convex entropy pair was left implicit in that work. We make this absence explicit and rigorous. Specifically, we derive the full compatibility system for the Hessian H = ∇²η of any candidate entropy η, and prove that positive definiteness of H is incompatible with the constraints imposed by the flux structure on any open domain containing states with both positive and negative velocity. The proof exploits an affine dependence of the (4,4)-entry of H on the velocity variable, which changes sign and therefore cannot be globally positive definite. As a direct consequence, the classical entropy-based convergence framework of Lax, Glimm, and DiPerna does not apply to this system in its full generality. In the second part of the paper, we carry out a complete wave-structure analysis: we compute all right eigenvectors of the flux Jacobian, classify the four characteristic fields as two linearly degenerate contact families and two nonlinear families, and identify an inflection locus {L = 3/5} inside the hyperbolicity domain on which genuine nonlinearity fails. The resulting wave pattern in the Riemann problem is a two-contact structure bracketed by two nonlinear waves, with composite waves appearing whenever the inflection locus is crossed. These results provide the analytical foundation for a companion numerical paper in which a Godunov-type scheme is constructed and the numerical viscosity is shown to serve as the admissibility mechanism replacing convex entropy.

Keywords : Hyperbolic conservation laws; biofilm model; convex entropy; entropy pair; Riemann problem; wave structure; genuine nonlinearity; linearly degenerate; mixture theory; vanishing viscosity.