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THE HYDROSTATIC STOKES SEMIGROUP AND WELL-POSEDNESS OF THE PRIMITIVE EQUATIONS ON SPACES OF BOUNDED FUNCTIONS

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Please use this identifier to cite or link to this item:http://doi.org/10.14943/81805

Title: THE HYDROSTATIC STOKES SEMIGROUP AND WELL-POSEDNESS OF THE PRIMITIVE EQUATIONS ON SPACES OF BOUNDED FUNCTIONS
Authors: Giga, Yoshikazu Browse this author
Gries, Mathis Browse this author
Hieber, Matthias Browse this author
Hussein, Amru Browse this author
Kashiwabara, Takahito Browse this author
Issue Date: 8-Feb-2018
Publisher: Department of Mathematics, Hokkaido University
Journal Title: Hokkaido University Preprint Series in Mathematics
Volume: 1109
Start Page: 1
End Page: 30
Abstract: Consider the 3-d primitive equations in a layer domain Ω = G×(−h,0), G = (0,1)2, subject to mixed Dirichlet and Neumann boundary conditions at z = −h and z = 0, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form a = a1 + a2, where a1 ∈ C(G;Lp(−h,0)), a2 ∈ L∞(G;Lp(−h,0)) for p > 3, and where a1 is periodic in the horizontal variables and a2 is sufficiently small. In particular, no differentiability condition on the data is assumed. The approach relies on L∞ H Lp z(Ω)-estimates for terms of the form t1/2∥∂zetAσPf∥L∞ H Lp z(Ω) ≤ Cetβ∥f∥L∞ H Lp z(Ω) for t > 0, where etAσ denotes the hydrostatic Stokes semigroup. The difficulty in proving estimates of this form is that the hydrostatic Helmholtz projection P fails to be bounded with respect to the L∞-norm. The global strong well-posedness result is then obtained by an iteration scheme, splitting the data into a smooth and a rough part and by combining a reference solution for smooth data with an evolution equation for the rough part.
Type: bulletin (article)
URI: http://hdl.handle.net/2115/68294
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

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