Global Strong Well-Posedness of the Three Dimensional Primitive Equations in $${L^p}$$ L p -Spaces

In this article, an Lp-approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data a∈[Xp,D(Ap)]1/p provided p∈[6/5,∞). To this end, the hydrostatic Stokes operator Ap defined o...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2016-09, Vol.221 (3), p.1077-1115
Main Authors: Hieber, Matthias, Kashiwabara, Takahito
Format: Article
Language:English
Subjects:
Mathematical analysis
Primitive equations
Well posed problems
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Summary:In this article, an Lp-approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data a∈[Xp,D(Ap)]1/p provided p∈[6/5,∞). To this end, the hydrostatic Stokes operator Ap defined on Xp, the subspace of Lp associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing p large, one obtains global well-posedness of the primitive equations for strong solutions for initial data a having less differentiability properties than H1, hereby generalizing in particular a result by Cao and Titi (Ann Math 166:245–267, 2007) to the case of non-smooth initial data.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-016-0979-x