Strong Solutions of the Equations for Viscoelastic Fluids in Some Classes of Large Data

TL;DR

This paper proves the global existence and uniqueness of strong solutions for incompressible viscoelastic fluids with large initial velocities under certain conditions, highlighting elasticity's role in preventing singularities similar to viscosity.

Contribution

It establishes the global well-posedness of strong solutions for viscoelastic fluids with large initial velocities and analyzes the asymptotic behavior of fluid particles.

Findings

Elasticity prevents singularity formation in large velocity flows.

Fluid particle lines tend to straighten over time.

Viscoelastic fluid motion approximates a linear pressureless flow asymptotically.

Abstract

We study the existence and uniqueness of global strong solutions to the equations of an incompressible viscoelastic fluid in a spatially periodic domain, and show that a unique strong solution exists globally in time if the initial deformation and velocity are small for the given physical parameters. In particular, the initial velocity can be large for the large elasticity coefficient. The result of this paper mathematically verifies that the elasticity can prevent the formation of singularities of strong solutions with large initial velocity, thus playing a similar role to viscosity in preventing the formation of singularities in viscous flows. Moreover, for given initial velocity perturbation and zero initial deformation around the rest state, we find, as the elasticity coefficient or time go to infinity, that (1) any straight line segment $l^0$ consisted of fluid particles in the rest state, after being bent by a velocity perturbation, will turn into a straight line segment that is parallel to $l^0$ and has the same length as $l^0$. (2) the motion of the viscoelastic fluid can be approximated by a linear pressureless motion in Lagrangian coordinates, even when the initial velocity is large. Moreover, the above mentioned phenomena can also be found in the corresponding compressible fluid case.