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May, 2021 Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces
Tsukasa IWABUCHI, Takayoshi OGAWA
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J. Math. Soc. Japan Advance Publication 1-42 (May, 2021). DOI: 10.2969/jmsj/81598159

Abstract

We consider the compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces $\dot{B}^{n/p}_{p,1} \times \dot{B}^{n/p-1}_{p,1}$ for all $1 \le p < 2n$. However, if the data is in a larger scaling invariant class such as $p > 2n$, then the system is not well-posed. In this paper, we demonstrate that for the critical case $p = 2n$ the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin and Haspot are indeed sharp in the framework of the homogeneous Besov spaces.

Funding Statement

The first author was supported by JSPS Grant-in-Aid for Young Scientists (A) (No. 17H04824). The second author was supported by JSPS Grant-in-Aid, Scientific Research (S) (No. 19H05597) and JSPS Challenging Research (Pioneering) (No. 17H06199).

Citation

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Tsukasa IWABUCHI. Takayoshi OGAWA. "Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces." J. Math. Soc. Japan Advance Publication 1 - 42, May, 2021. https://doi.org/10.2969/jmsj/81598159

Information

Received: 29 October 2018; Revised: 2 August 2020; Published: May, 2021
First available in Project Euclid: 26 May 2021

Digital Object Identifier: 10.2969/jmsj/81598159

Subjects:
Primary: 35Q30
Secondary: 47J06, 76D05, 76N10

Rights: Copyright ©2021 Mathematical Society of Japan

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