NavierStokes Equations in a Rotating Frame in ${\mathbb R}^3$ with Initial Data Nondecreasing at Infinity
Preprint Series # 664 Giga, Yoshikazu and Inui, Katsuya and Mahalov, Alex and Matsui, Shin'ya NavierStokes Equations in a Rotating Frame in ${\mathbb R}^3$ with Initial Data Nondecreasing at Infinity. (2004); AbstractThreedimensional 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
$\tilde{\mbox{\bf{V}}}_0(y) \in L^{\infty}_{\sigma,a}({\mathbb R}^3)$.
Item Type:  Preprint 

Additional Information:  60 copies needed 

Uncontrolled Keywords:  Rotating NavierStokes equations, nondecreasing initial data, homogeneous Besov spaces, Riesz operators.


Subjects:  35xx PARTIAL DIFFERENTIAL EQUATIONS 

ID Code:  431 

