scholarly journals Non-uniform Dependence on Initial Data for the Camassa–Holm Equation in the Critical Besov Space

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Jinlu Li ◽  
Xing Wu ◽  
Yanghai Yu ◽  
Weipeng Zhu
Keyword(s):  
Initial Data ◽  
Besov Space ◽  
Holm Equation ◽  
Critical Besov Space

The dependences on initial data for the b-family equation in critical Besov space

2014 ◽  
Vol 177 (3) ◽  
pp. 471-492 ◽  
Author(s):  
Hao Tang ◽  
Shijie Shi ◽  
Zhengrong Liu
Keyword(s):  
Initial Data ◽  
Besov Space ◽  
Critical Besov Space

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.

scholarly journals The Cauchy problem for the generalized Camassa–Holm equation in Besov space

2014 ◽  
Vol 256 (8) ◽  
pp. 2876-2901 ◽  
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Yimin Zhang
Keyword(s):  
Cauchy Problem ◽  
Besov Space ◽  
Holm Equation ◽  
The Cauchy Problem

Global Existence for the Two-Dimensional Euler Equations in a Critical Besov Space

2011 ◽  
Vol 271-273 ◽  
pp. 791-796
Author(s):  
Kun Qu ◽  
Yue Zhang
Keyword(s):  
Global Existence ◽  
Transport Equation ◽  
Besov Space ◽  
Euler Equations ◽  
Existence Result ◽  
Two Dimensional ◽  
Critical Besov Space ◽  
Global Existence Result

In this paper we prove the global existence for the two-dimensional Euler equations in the critical Besov space. Making use of a new estimate of transport equation and Littlewood-Paley theory, we get the global existence result.

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