Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces

Xinghong Pan; Jiang Xu

doi:10.3934/dcds.2019085

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2019, Volume 39, Issue 4: 2021-2057. Doi: 10.3934/dcds.2019085

This issue Previous Article Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation Next Article Q-entropy for general topological dynamical systems

Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces

Xinghong Pan and
Jiang Xu^,

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received: March 2018

Revised: August 2018

Published: January 2019

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Abstract

In this paper, we are concerned with the compressible viscoelastic flows in whole space $ \mathbb{R}^n $ with $ n\geq2 $. We aim at extending the global existence in energy spaces (see [18] by Hu & Wang and [30] by Qian & Zhang) such that it holds in more general $ L^p $ critical spaces, which allows to the case of large highly oscillating initial velocity. Precisely, We define "two effective velocities" which are used to eliminate the coupling between the density, velocity and deformation tensor. Consequently, the global existence in the $ L^p $ critical framework is constructed by elementary energy approaches. In addition, the optimal time-decay estimates of strong solutions are firstly shown in the $ L^p $ framework, which improve recent decay efforts for compressible viscoelastic flows.

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

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