Ill-posedness of the basic equations of fluid dynamics in Besov spaces
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- by A. Cheskidov and R. Shvydkoy PDF
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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 $\|u_0\|_{B^{-1}_{\infty ,\infty }}$ 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.References
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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
- MR Author ID: 680409
- ORCID: 0000-0002-2589-2047
- 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2566571