Global Solutions for Incompressible Viscoelastic Fluids

TL;DR

This paper proves the existence of local and global smooth solutions for incompressible viscoelastic fluids modeled by Oldroyd-B equations, applicable in multiple dimensions and for various elastic complex fluids.

Contribution

It establishes the existence of solutions for the Oldroyd-B system in both local and global contexts, extending to multiple dimensions and related elastic fluid models.

Findings

Existence of local and global smooth solutions in 2D and 3D.

Results applicable to a broad class of elastic complex fluids.

Methods valid for both whole space and periodic boundary conditions.

Abstract

We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near equilibrium initial data. The results hold in both two and three dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals and mixture problems.