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

Abstract

We investigate global strong solutions for the incompressible viscoelastic system 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 for the incompressible Navier-Stokes equations, and one more derivative is needed for the deformation tensor. Like the classical incompressible Navier-Stokes, 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. *

Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism

“…Besides, we would like to mention that global regularity of solutions to 2D Oldroyd-B model with diffusive stress was obtained by Constantin and Kliegl [5]. And some interesting results for related Oldroyd type models of viscoelastic fluids can be found in [19,18,16,17,20,4,23,24]. We shall review these results later in Section 2.…”

“…Qian and Zhang [23] generalized the results to compressible case. The result of critical L p framework was given by Zhang and Fang [24].…”

Global Solutions to the Oldroyd-B Model with a Class of Large Initial Data

“…Besides, we would like to mention that Constantin and Kliegl [9] proved the global regularity of solutions in two dimensional case for the Oldroyd-B fluids with diffusive stress. An approach based on Lagrangian particle dynamics can be found in [19,20,21,23,24,25,30,32].…”

On the Cauchy problems for certain regularized models to the incompressible viscoelastic flow

“…Further discussions of these topics can be referred to previous studies. [9][10][11][12][13][14][15][16][17][18][19][20][21] A summary on some of the recent results can be found in a review paper. 22 We now revisit the system (1).…”