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Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

Published online by Cambridge University Press:  15 April 2002

Anatoli Babin ,
Alex Mahalov  and
Basil Nicolaenko
Anatoli Babin
Affiliation:
Department of Mathematics, University of California, Irvine, CA, 92697, USA.
Alex Mahalov
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA.
Basil Nicolaenko
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA.
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Abstract

Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D "primitive" Navier-Stokes equations for strong stratification (large stratification parameter N). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear "$2\frac{1}{2}$ dimensional" limit equations for N → +∞; smoothness assumptions are the same as for local existence theorems, that is initial data in Hα, α ≥ 3/4. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonant equations and convergence theorems.

Keywords

Fast singular oscillating limitsthree-dimensional Navier-Stokes equationsprimitive equations for geophysical fluid flows.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 2: Special issue for R. Teman's 60th birthday , March 2000 , pp. 201 - 222
Copyright
© EDP Sciences, SMAI, 2000

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