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

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