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THE HYDROSTATIC STOKES SEMIGROUP AND WELLPOSEDNESS OF THE PRIMITIVE EQUATIONS ON SPACES OF BOUNDED FUNCTIONS
Title:  THE HYDROSTATIC STOKES SEMIGROUP AND WELLPOSEDNESS 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:  8Feb2018 
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 3d 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 wellposed 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 wellposedness 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

Submitter: 数学紀要登録作業用
