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

Xianpeng Hu; Hao Wu

doi:10.3934/dcds.2015.35.3437

Article Contents

2015, Volume 35, Issue 8: 3437-3461. Doi: 10.3934/dcds.2015.35.3437

This issue Previous Article Emergence of phase-locked states for the Winfree model in a large coupling regime Next Article Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains

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

Xianpeng Hu^1, and
Hao Wu^2,

1.
Department of Mathematics, City University of Hong Kong, Hong Kong
2.
School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, 200433 Shanghai

Received: August 31, 2014

Revised: October 31, 2014

Published: May 2017

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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.

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Mathematics Subject Classification: Primary: 35B40, 35Q35; Secondary: 35L60.

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