208
for all z in (1 - Pn.)D(A.B).
Then it is easy to see that (1.12) and (1.13) are satisfied with n = k and
and kl = k2 = c'.
Ak = IIn.+1 - 2cOII~.+I'
Ak = lin. + 2cOII~.,
(4.26)
All the other hypotheses (1.7)-(1.11) follow promptly j (1.7) is satisfied for n sufficiently large, n ;:: nl, which is not restrictive at all.
Then Theorem 2.1 applies without any restriction (except to require that n ;:: nt).
Furthermore, in view of (4.8), we show that
° E M and M is tangent at ° to the space P E,
(4.27)
a property which appears naturally in the context of slow manifolds.
Remark 4.1. For more details on the relations with slow manifolds, the reader is refereed
to A. Debussche and R. Temam (1991a,b) and R. Temam (1990).
4.3. An Equation of Navier-Stokes Type
In this section we give an application of Theorem 2.1 in both the self-adjoint and
nonself-adjoint cases. We present the equations in the non prepared case j it is understood that the equations must be prepared.
We consider, in space dimension l ;:: 2, the Navier-Stokes equations with a higherorder viscosity term
au at +c(-~ru-v~u+(u·V)u+Vp=f,
(4.28)
v . u = 0,
(4.29)
The functions u = u( x, t) and p = p( x, t) are defined on JR t X ~, taking respectively values in JRI and JR, u = (UI, ... , Ut)j c and v are strictly positive numbers, r > 1.
Of course (4.28) reduces to the usual Navier-Stokes equations when c = 0, but here we
assume c > 0.
We restrict ourselves to the space-periodic case although other boundary conditions
can be considered. We assume that U and p are periodic in each direction Xl, ••• , Xl with
period L > 0,
{
u(x + Lei, t) = u(x, t),
p(x + Lei, t) = p(x, t),
i = 1, ... ,l,
where {el' ... , et} is the natural basis of JR t . Furthermore we assume that
In f(x)dx = 0, In u(x,t)dx = 0, In p(x,t)dx = 0,
(4.30)
(4.31 )
for all z in (1 - Pn.)D(A.B).
Then it is easy to see that (1.12) and (1.13) are satisfied with n = k and
and kl = k2 = c'.
Ak = IIn.+1 - 2cOII~.+I'
Ak = lin. + 2cOII~.,
(4.26)
All the other hypotheses (1.7)-(1.11) follow promptly j (1.7) is satisfied for n sufficiently large, n ;:: nl, which is not restrictive at all.
Then Theorem 2.1 applies without any restriction (except to require that n ;:: nt).
Furthermore, in view of (4.8), we show that
° E M and M is tangent at ° to the space P E,
(4.27)
a property which appears naturally in the context of slow manifolds.
Remark 4.1. For more details on the relations with slow manifolds, the reader is refereed
to A. Debussche and R. Temam (1991a,b) and R. Temam (1990).
4.3. An Equation of Navier-Stokes Type
In this section we give an application of Theorem 2.1 in both the self-adjoint and
nonself-adjoint cases. We present the equations in the non prepared case j it is understood that the equations must be prepared.
We consider, in space dimension l ;:: 2, the Navier-Stokes equations with a higherorder viscosity term
au at +c(-~ru-v~u+(u·V)u+Vp=f,
(4.28)
v . u = 0,
(4.29)
The functions u = u( x, t) and p = p( x, t) are defined on JR t X ~, taking respectively values in JRI and JR, u = (UI, ... , Ut)j c and v are strictly positive numbers, r > 1.
Of course (4.28) reduces to the usual Navier-Stokes equations when c = 0, but here we
assume c > 0.
We restrict ourselves to the space-periodic case although other boundary conditions
can be considered. We assume that U and p are periodic in each direction Xl, ••• , Xl with
period L > 0,
{
u(x + Lei, t) = u(x, t),
p(x + Lei, t) = p(x, t),
i = 1, ... ,l,
where {el' ... , et} is the natural basis of JR t . Furthermore we assume that
In f(x)dx = 0, In u(x,t)dx = 0, In p(x,t)dx = 0,
(4.30)
(4.31 )