TY - JOUR

T1 - Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces

AU - Iwabuchi, Tsukasa

AU - Ogawa, Takayoshi

N1 - Funding Information:
2010 Mathematics Subject Classification. Primary 35Q30; Secondary 76D05, 76N10, 47J06. Key Words and Phrases. the compressible Navier–Stokes system, ill-posedness, critical Besov spaces. 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).
Publisher Copyright:
© 2022 The Mathematical Society of Japan

PY - 2022

Y1 - 2022

N2 - 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 B pn/p1 ×B pn/p1−1 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.

AB - 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 B pn/p1 ×B pn/p1−1 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.

KW - critical Besov spaces

KW - ill-posedness

KW - the compressible Navier–Stokes system

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U2 - 10.2969/JMSJ/81598159

DO - 10.2969/JMSJ/81598159

M3 - Article

AN - SCOPUS:85129938321

SN - 0025-5645

VL - 74

SP - 353

EP - 394

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

IS - 2

ER -