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

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Abstract

In this article, an \({L^p}\)-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 \({a \in [X_p,D(A_p)]_{1/p}}\) provided \({p \in [6/5,\infty)}\). To this end, the hydrostatic Stokes operator \({A_p}\) defined on \({X_p}\), the subspace of \({L^p}\) 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 \({H^1}\), hereby generalizing in particular a result by Cao and Titi (Ann Math 166:245–267, 2007) to the case of non-smooth initial data.

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Authors and Affiliations

  1. Department of Mathematics, TU Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Matthias Hieber

  2. Benedum Engineering Hall, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Matthias Hieber

  3. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oklahoma, Meguro, Tokyo, 152-8511, Japan

    Takahito Kashiwabara

Authors
  1. Matthias Hieber
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  2. Takahito Kashiwabara
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Correspondence to Matthias Hieber.

Additional information

Communicated by P. Constantin

This work was supported by the DFG-JSPS International Research Training Group 1529 on Mathematical Fluid Dynamics.

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Hieber, M., Kashiwabara, T. Global Strong Well-Posedness of the Three Dimensional Primitive Equations in \({L^p}\)-Spaces. Arch Rational Mech Anal 221, 1077–1115 (2016). https://doi.org/10.1007/s00205-016-0979-x

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  • DOI: https://doi.org/10.1007/s00205-016-0979-x

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