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Ill-posedness for the Navier-Stokes equa...
Wang, Baoxiang...

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Ill-posedness for the Navier-Stokes equations in critical Besov spaces $\dot B^{-1}_{\infty,q}$ by Wang, Baoxiang ( Author )
Lemonjar Open Access
Australian National University
04-08-2023
We study the Cauchy problem for the incompressible Navier-Stokes equation \begin{align} u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= \delta u_0. \label{NS} \end{align} For arbitrarily small $\delta>0$, we show that the solution map $\delta u_0 \to u$ in critical Besov spaces $\dot B^{-1}_{\infty,q}$ ($\forall \ q\in [1,2]$) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in $BMO^{-1}$. Taking notice of the embedding $\dot B^{-1}_{\infty,q} \subset BMO^{-1}$ ($q\le 2$), we see that for sufficiently small $\delta>0$, $u_0\in \dot B^{-1}_{\infty,q} $ ($q\le 2$) can guarantee that the Navier-Stokes equation has a unique global solution in $BMO^{-1}$, however, this solution is instable in $ \dot B^{-1}_{\infty,q} $ and the solution can have an inflation in $\dot B^{-1}_{\infty,q}$ for certain initial data. So, our result indicates that two different topological structures in the same space may determine the well and ill posedness, respectively.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1403.2461
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