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arXiv · 2604.13330

Derivation of effective kinetic equations describing oscillations in viscoelasticity and in compressible Navier-Stokes

Athanasios E. Tzavaras
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Abstract

These lecture notes are devoted to solutions of hyperbolic-parabolic systems with persistent oscillations. We consider two examples both from mechanics: (i) The system of viscoelasticity of Kelvin-Voigt type with strain energies involving double well potentials, as employed in phase transitions. (ii) The compressible Navier-Stokes equations for a barotropic gas. For each system we construct solutions with persistent oscillations. In a later part we consider the nonlinear homogenization problem. For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids we show how ideas from the kinetic formulation of conservation laws can be used to derive effective equations. The effective equation consists by a kinetic equation coupled with the macroscopic flow.

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Athanasios E. Tzavaras. 2026-04-14. Derivation of effective kinetic equations describing oscillations in viscoelasticity and in compressible Navier-Stokes. https://arxiv.org/abs/2604.13330

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