Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces
- 21 April 2022
- journal article
- research article
- Published by Mathematical Society of Japan (Project Euclid) in Journal of the Mathematical Society of Japan
- Vol. 74 (2), 1-42
- https://doi.org/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.
Keywords
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