Ill-posedness of the Navier-Stokes equations in a critical space in 3D
Abstract
We prove that the Cauchy problem for the three dimensional Navier-Stokes equations is ill posed in $\dot{B}^{-1,\infty}_{\infty}$ in the sense that a ``norm inflation'' happens in finite time. More precisely, we show that initial data in the Schwartz class $\mathcal{S}$ that are arbitrarily small in $\dot{B}^{-1, \infty}_{\infty}$ can produce solutions arbitrarily large in $\dot{B}^{-1, \infty}_{\infty}$ after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in $\dot{B}^{-1, \infty}_{\infty}$ at the origin.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2008
- DOI:
- 10.48550/arXiv.0807.0882
- arXiv:
- arXiv:0807.0882
- Bibcode:
- 2008arXiv0807.0882B
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 16 pages, no figures