Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces
Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces
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DOI：
10.2969/jmsj/81598159
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发表时间：
2021-05
影响因子：
0.7
通讯作者：
T. Iwabuchi;T. Ogawa
中科院分区：
文献类型：
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作者：
T. Iwabuchi;T. Ogawa
我们考虑临界Besov空间中的可压缩Navier-Stokes方程组。已知系统在齐次Besov空间<$n p p，1 × <$n p −1 p，1的尺度半不变空间中是（半）适定的，对所有1 ≤ p 2n，则系统不是适定的。在本文中，我们证明了临界情况下p = 2n的系统是不适定的，通过显示的初始数据序列的构造，以显示在临界空间的解映射的不连续性。我们的结果表明Danchin [10]和Haspot [18]的适定性结果在齐次Besov空间的框架下确实是尖锐的。
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 Ḃ n p p,1 × Ḃ n p −1 p,1 for all 1 ≤ 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 [10] and Haspot [18] are indeed sharp in the framework of the homogeneous Besov spaces.