# On dissipative solutions to a system arising in viscoelasticity

## TL;DR

This paper introduces the concept of dissipative solutions for a viscoelastic fluid model, proving their global existence and uniqueness with strong solutions when they exist, advancing mathematical understanding of such systems.

## Contribution

It presents a new framework of dissipative solutions for viscoelasticity equations, establishing their global existence and uniqueness properties.

## Key findings

- Dissipative solutions exist globally for finite energy initial data.
- Dissipative and strong solutions coincide when the latter exist.
- The notion of dissipative solutions provides a robust analytical tool.

## Abstract

We consider a model for an incompressible visoelastic fluid. It consists of the Navier-Stokes equations involving an elastic term in the stress tensor and a transport equation for the evolution of the deformation gradient. The novel feature of the paper is the introduction of the notion of a dissipative solution and its analysis. We show that dissipative solutions exist globally in time for arbitrary finite energy initial data and that a dissipative solution and a strong solution emanating from the same initial data coincide as long as the latter exists.