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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Improved a priori bounds for thermal fluid equations


Author: Andrei Tarfulea
Journal: Trans. Amer. Math. Soc. 371 (2019), 2719-2737
MSC (2010): Primary 76N10; Secondary 35A01, 35A23, 35Q35
DOI: https://doi.org/10.1090/tran/7529
Published electronically: September 18, 2018
MathSciNet review: 3896095
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Abstract: We consider two hydrodynamic model problems (one incompressible and one compressible) with three-dimensional fluid flow on the torus and temperature-dependent viscosity and conductivity. The ambient heat for the fluid is transported by the flow and fed by the local energy dissipation, modeling the transfer of kinetic energy into thermal energy through fluid friction. Both the viscosity and conductivity grow with the local temperature. We prove a strong a priori bound on the enstrophy of the velocity weighed against the temperature for initial data of arbitrary size, requiring only that the conductivity be proportionately larger than the viscosity (and, in the incompressible case, a bound on the temperature as a Muckenhoupt weight).


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Additional Information

Andrei Tarfulea
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois
Email: atarfulea@math.uchicago.edu

DOI: https://doi.org/10.1090/tran/7529
Received by editor(s): February 13, 2017
Received by editor(s) in revised form: February 27, 2017, and January 3, 2018
Published electronically: September 18, 2018
Additional Notes: The author was partially supported by NSF grant DMS-1246999.
Article copyright: © Copyright 2018 American Mathematical Society