Regularity and Blow up for Active Scalars

A. Kiselev

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 225-255
  • ISSN: 0973-5348

Abstract

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We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.

How to cite

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Kiselev, A.. "Regularity and Blow up for Active Scalars." Mathematical Modelling of Natural Phenomena 5.4 (2010): 225-255. <http://eudml.org/doc/197710>.

@article{Kiselev2010,
abstract = {We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.},
author = {Kiselev, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {active scalars; global regularity; finite time blow up; nonlocal maximum principle},
language = {eng},
month = {5},
number = {4},
pages = {225-255},
publisher = {EDP Sciences},
title = {Regularity and Blow up for Active Scalars},
url = {http://eudml.org/doc/197710},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Kiselev, A.
TI - Regularity and Blow up for Active Scalars
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 225
EP - 255
AB - We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.
LA - eng
KW - active scalars; global regularity; finite time blow up; nonlocal maximum principle
UR - http://eudml.org/doc/197710
ER -

References

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