On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces

Qionglei Chen ^[1] ; Changxing Miao ^[1] ; Zhifei Zhang ^[2]
1. [1] Institute of Applied Physics and Computational Mathematics
  
  Institute of Applied Physics and Computational Mathematics
  
  China
2. [2] Peking University
  
  Peking University
  
  China
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 4, 2015, págs. 1375-1402
Idioma: inglés
DOI: 10.4171/RMI/872
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Resumen
- We prove the ill-posedness of the 3-D baratropic Navier–Stokes equation for the initial density and velocity belonging to the critical Besov space (B˙3/p,1+ρ¯,B˙3/p−1p,1)(B˙p,13/p+ρ¯,B˙p,13/p−1) for p>6 in the sense that a "norm inflation" happens in finite time, here ρ¯ is a positive constant. While, the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space (B˙3/p,1+ρ¯,B˙3/p−1p,1,B˙3/p−2p,1)(B˙p,13/p+ρ¯,B˙p,13/p−1,B˙p,13/p−2) for p>3.