On a transport equation with nonlocal drift
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- by Luis Silvestre and Vlad Vicol PDF
- Trans. Amer. Math. Soc. 368 (2016), 6159-6188 Request permission
Abstract:
In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \[ \partial _t \theta +u \; \partial _x \theta = 0, \qquad u = H \theta , \] where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion \[ \partial _t \theta + u \; \partial _x \theta + \Lambda ^\gamma \theta = 0, \qquad u = H \theta , \] where $\Lambda = (-\Delta )^{1/2}$, and $1/2 \leq \gamma <1$. Our results also apply to the model with velocity field $u = \Lambda ^s H \theta$, where $s \in (-1,1)$. We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in $C^{(s+1)/2}$, for all positive time.References
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Additional Information
- Luis Silvestre
- Affiliation: Department of Mathematics, The University of Chicago, Chicago, Illinois 60637
- MR Author ID: 757280
- Email: luis@math.uchicago.edu
- Vlad Vicol
- Affiliation: Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, New Jersey 08544
- MR Author ID: 846012
- ORCID: setImmediate$0.00243841196800898$2
- Email: vvicol@math.princeton.edu
- Received by editor(s): August 5, 2014
- Published electronically: November 6, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6159-6188
- MSC (2010): Primary 35Q35
- DOI: https://doi.org/10.1090/tran6651
- MathSciNet review: 3461030