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On a transport equation with nonlocal drift


Authors: Luis Silvestre and Vlad Vicol
Journal: Trans. Amer. Math. Soc. 368 (2016), 6159-6188
MSC (2010): Primary 35Q35
DOI: https://doi.org/10.1090/tran6651
Published electronically: November 6, 2015
MathSciNet review: 3461030
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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

$\displaystyle \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

$\displaystyle \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.

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Additional Information

Luis Silvestre
Affiliation: Department of Mathematics, The University of Chicago, Chicago, Illinois 60637
Email: luis@math.uchicago.edu

Vlad Vicol
Affiliation: Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, New Jersey 08544
Email: vvicol@math.princeton.edu

DOI: https://doi.org/10.1090/tran6651
Received by editor(s): August 5, 2014
Published electronically: November 6, 2015
Article copyright: © Copyright 2015 American Mathematical Society