Global Existence of Strong Solution for Equations Related to the Incompressible Viscoelastic Fluids in the Critical $L^p$ Framework

Abstract

We investigate the global strong solutions for a system of equations related to the incompressible viscoelastic fluids of Oldroyd--B type with the initial data close to a stable equilibrium. We obtain the existence and uniqueness of the global solution in a functional setting invariant by the scaling of the associated equations, where the initial velocity has the same critical regularity index as the incompressible Navier--Stokes equations, and one more derivative is needed for the deformation tensor. Like the classical incompressible Navier--Stokes equations, one may construct the unique global solution for a class of large highly oscillating initial velocity. Our result also implies that the deformation tensor $F$ has the same regularity as the density of the compressible Navier--Stokes equations.