Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows

Abstract

We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space ${\mathbb{R}}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of global smooth solutions near equilibrium. Then under additional assumptions that the initial data belong to $L^1$ and their Fourier modes do not degenerate at low frequencies, we obtain the optimal $L^2$ decay rates for the global smooth solutions and their spatial derivatives. At last, we establish the weak-strong uniqueness property in the class of finite energy weak solutions for the incompressible viscoelastic system.

28 pages, finished in 2012, accepted by DCDS-A in 2014

References in corpus (2)

• Global Solutions for Incompressible Viscoelastic Fluids
• A blowup criterion for ideal viscoelastic flow

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• On a diffuse interface model for incompressible viscoelastic two-phase flows
• On the energy equality for the 3D incompressible viscoelastic flows
• Forward self-similar solutions to the viscoelastic Navier-Stokes equation with damping
• The Cauchy problem for an inviscid Oldroyd-B model in ${\mathbb{R}}^3$
• Strong Solutions of the Equations for Viscoelastic Fluids in Some Classes of Large Data