2022 年 74 巻 2 号 p. 353-394
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/𝑝𝑝,1 × \dot{𝐵}𝑛/𝑝−1𝑝,1 for all 1 ≤ 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.
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