# Symmetric forms for hyperbolic-parabolic systems of multi-gradient fluids

## TL;DR

This paper develops symmetric formulations for hyperbolic-parabolic multi-gradient fluid systems, ensuring thermodynamic consistency, stability near equilibrium, and a divergence form for equations, advancing the mathematical understanding of complex fluid models.

## Contribution

It introduces symmetric-Hermitian and divergence form equations for multi-gradient fluids, linking thermodynamics with dispersive and stability analysis.

## Key findings

- Equations are compatible with thermodynamics laws.
- Equilibrium positions are proven stable under convex energy conditions.
- System can be expressed in divergence form and as symmetric-Hermitian near equilibrium.

## Abstract

We consider multi-gradient fluids endowed with a volumetric internal energy which is a function of mass density, volumetric entropy and their successive gradients. We obtained the thermodynamic forms of equation of motions and equation of energy, and the motions are compatible with the two laws of thermodynamics. The equations of multi-gradient fluids belong to the class of dispersive systems. In the conservative case, we can replace the set of equations by a quasi-linear system written in a divergence form. Near an equilibrium position, we obtain a symmetric-Hermitian system of equations in the form of Godunov's systems. The equilibrium positions are proved to be stable when the total volume energy of the fluids is a convex function with respect to convenient conjugated variables-called main field-of mass density, volumetric entropy, their successive gradients, and velocity.