Relative entropy in diffusive relaxation

TL;DR

This paper proves convergence of entropy weak solutions from compressible gas dynamics with friction and viscoelasticity to simpler models using a relative entropy approach, highlighting the method's effectiveness in complex systems.

Contribution

It introduces a novel application of the relative entropy method to establish convergence in diffusive limits for both gas dynamics with friction and viscoelasticity with memory.

Findings

Convergence from gas dynamics to porous media equation established

Convergence from viscoelasticity with memory to rate-type system demonstrated

Relative entropy functional effectively measures solution proximity

Abstract

We establish convergence in the diffusive limit from entropy weak solutions of the equations of compressible gas dynamics with friction to the porous media equation away from vacuum. The result is based on a Lyapunov type of functional provided by a calculation of the relative entropy. The relative entropy method is also employed to establish convergence from entropic weak solutions of viscoelasticity with memory to the system of viscoelasticity of the rate-type.