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The Cauchy problem for an inviscid Oldroyd-B model in three dimensions: global well posedness and optimal decay rates

Part of: Dynamical problems Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  09 February 2022

Sili Liu ,
Wenjun Wang  and
Huanyao Wen
Sili Liu
Affiliation:
School of Mathematics, South China University of Technology, Guangzhou 510631, China maslliu@mail.scut.edu.cn
Wenjun Wang
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China wwj001373@hotmail.com
Huanyao Wen*
Affiliation:
School of Mathematics, South China University of Technology, Guangzhou 510631, China mahywen@scut.edu.cn
*
*Corresponding author.
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Abstract

In this paper, we consider the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with [47] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. Moreover, we prove that the decay rates for the highest-order derivatives of the solutions are optimal, which is of independent interest. Our proof relies on Fourier theory and delicate energy method.

Keywords

Inviscid compressible Oldroyd-B modelglobal well posednessdecay rates

MSC classification

Primary: 76N10: Existence, uniqueness, and regularity theory
Secondary: 76N17: Viscous-inviscid interaction 74H40: Long-time behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 2 , April 2023 , pp. 441 - 490
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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