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

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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: February 28, 2018

Revised: July 31, 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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