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An inviscid regularization for the surface quasi‐geostrophic equation
Author(s) -
Khouider Boualem,
Titi Edriss S.
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20218
Subject(s) - inviscid flow , regularization (linguistics) , mathematics , singularity , mathematical analysis , classical mechanics , physics , computer science , artificial intelligence
Abstract Inspired by recent developments in Berdina‐like models for turbulence, we propose an inviscid regularization for the surface quasi‐geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter α tends to 0 for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite‐time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. © 2007 Wiley Periodicals, Inc.