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ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN $L^{6}$ FRAMEWORK

Part of: Equations of mathematical physics and other areas of application Parabolic equations and systems

Published online by Cambridge University Press:  29 June 2017

Jiecheng Chen  and
Renhui Wan
Jiecheng Chen
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China (jcchen@zjnu.edu.cn)
Renhui Wan
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China (rhwanmath@163.com; rhwanmath@zju.edu.cn)
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Abstract

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$$(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$
To accomplish this goal, we require a norm inflation coming from the coupling term $L(a)\unicode[STIX]{x1D6E5}u$ instead of $u\cdot \unicode[STIX]{x1D6FB}u$ and construct a new decomposition of the density.

Keywords

Navier–Stokes equationsill-posednessBesov space

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 4 , July 2019 , pp. 829 - 854
Copyright
© Cambridge University Press 2017 

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Footnotes

The original version of this article was submitted without an identified corresponding author. A notice detailing this has been published and the error rectified in the online PDF and HTML copies.

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Author for correspondence.

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