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Ill-posedness of the basic equations of fluid dynamics in Besov spaces


Authors: A. Cheskidov and R. Shvydkoy
Journal: Proc. Amer. Math. Soc. 138 (2010), 1059-1067
MSC (2000): Primary 76D03; Secondary 35Q30
DOI: https://doi.org/10.1090/S0002-9939-09-10141-7
Published electronically: October 22, 2009
MathSciNet review: 2566571
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Abstract: We give a construction of a divergence-free vector field $ u_0 \in H^s \cap B^{-1}_{\infty,\infty}$, for all $ s<1/2$, with arbitrarily small norm $ \Vert u_0\Vert _{B^{-1}_{\infty,\infty}}$ such that any Leray-Hopf solution to the Navier-Stokes equation starting from $ u_0$ is discontinuous at $ t=0$ in the metric of $ B^{-1}_{\infty,\infty}$. For the Euler equation a similar result is proved in all Besov spaces $ B^s_{r,\infty}$ where $ s>0$ if $ r>2$, and $ s>n(2/r-1)$ if $ 1 \leq r \leq 2$. This includes the space $ B^{1/3}_{3,\infty}$, which is known to be critical for the energy conservation in ideal fluids.


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Additional Information

A. Cheskidov
Affiliation: Department of Mathematics, Statistics and Computer Science, M/C 249, University of Illinois, Chicago, Illinois 60607
Email: acheskid@math.uic.edu

R. Shvydkoy
Affiliation: Department of Mathematics, Statistics and Computer Science, M/C 249, University of Illinois, Chicago, Illinois 60607
Email: shvydkoy@math.uic.edu

DOI: https://doi.org/10.1090/S0002-9939-09-10141-7
Keywords: Euler equation, Navier-Stokes equation, ill-posedness, Besov spaces
Received by editor(s): April 20, 2009
Received by editor(s) in revised form: July 22, 2009
Published electronically: October 22, 2009
Additional Notes: The work of the first author is partially supported by NSF grant DMS-0807827
The work of the second author is partially supported by NSF grant DMS-0907812 and CRDF grant RUM1-2842-RO-06
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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