A class of global large solutions to the compressible Navier–Stokes–Korteweg system in critical Besov spaces

2020 ◽  
Vol 20 (4) ◽  
pp. 1531-1561 ◽  
Author(s):  
Shunhang Zhang
Keyword(s):  
Besov Spaces ◽  
Navier Stokes ◽  
Large Solutions ◽  
Critical Besov Spaces

ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN FRAMEWORK

2017 ◽  
Vol 18 (4) ◽  
pp. 829-854 ◽  
Author(s):  
Jiecheng Chen ◽  
Renhui Wan
Keyword(s):  
Initial Data ◽  
Besov Space ◽  
Besov Spaces ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Coupling Term ◽  
Navier Stokes Equations ◽  
Critical Besov Space ◽  
Image Position ◽  
Critical Besov Spaces

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$$(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying $$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$ To accomplish this goal, we require a norm inflation coming from the coupling term $L(a)\unicode[STIX]{x1D6E5}u$ instead of $u\cdot \unicode[STIX]{x1D6FB}u$ and construct a new decomposition of the density.

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