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NavierStokes Equations in a Rotating Frame in R 3 with Initial Data Nondecreasing at Infinity
Title:  NavierStokes Equations in a Rotating Frame in R 3 with Initial Data Nondecreasing at Infinity 
Authors:  Giga, Yoshikazu Browse this author  Inui, Katsuya Browse this author  Mahalov, Alex Browse this author  Matsui, Shin'ya Browse this author 
Issue Date:  2004 
Publisher:  Department of Mathematics, Hokkaido University 
Journal Title:  Hokkaido University Preprint Series in Mathematics 
Volume:  664 
Start Page:  1 
End Page:  28 
Abstract:  Threedimensional rotating NavierStokes equations are considered with a constant Coriolis parameter $\Omega$ and initial data nondecreasing at infinity. In contrast to the nonrotating case ($\Omega=0$), it is shown for the problem with rotation ($\Omega \neq 0$) that Green's function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to $L^1({\mathbb R}^3)$. Moreover, the corresponding integral operator is unbounded in the space $L^{\infty}_{\sigma}({\mathbb R}^3)$ of solenoidal vector fields in ${\mathbb R}^3$ and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in $L^{\infty}_{\sigma}({\mathbb R}^3)$. Local in time, uniform in $\Omega$ unique solvability of the rotating NavierStokes equations is proven for initial velocity fields in the space $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which consists of $L^{\infty}$ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space ${\dot B}_{\infty,1}^0$ which is smaller than $L^\infty$ but still contains various periodic and almost periodic functions. This restriction of initial data to $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which is a subspace of $L^{\infty}_{\sigma}({\mathbb R}^3)$ is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. The proof of uniform in $\Omega$ local in time unique solvability requires detailed study of the symbol of this semigroup and obtaining uniform in $\Omega$ estimates of the corresponding operator norms in Banach spaces. Using the rotation transformation, we also obtain local in time, uniform in $\Omega$ solvability of the classical 3D NavierStokes equations in ${\mathbb R}^3$ with initial velocity and vorticity of the form $\mbox{\bf{V}}(0)=\tilde{\mbox{\bf{V}}}_0(y) + \frac{\Omega}{2} e_3 \times y$, $\mbox{curl} \mbox{\bf{V}}(0)=\mbox{curl} \tilde{\mbox{\bf{V}}}_0(y) + \Omega e_3$ where $\tilde{\mbox{\bf{V}}}_0(y) \in L^{\infty}_{\sigma,a}({\mathbb R}^3)$. 
Type:  bulletin (article) 
URI:  http://hdl.handle.net/2115/69469 
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
