210
(and (4.31) is satisfied). The existence of u,p, follows e.g. from J.L. Lions (1969), R.
Temam (1977). Let u be any such solution and set v = u - u and q = p - p. We have
av at + e( -~tv - v~v + (u. V)v + (v· V)u + (v· V)v + Vq = 0,
V·v=O,
v(x+Le;,t)=v(x,t), q(x,Le;,t)=q(x,t), i=I, ... ,f.
(4.36)
(4.37)
(4.38)
Equations (4.36)-(4.38) amount to an equation for v of type (4.4). Here we have
D(A) = D(Ao) and
Av = e( -~tv + II«u· V)v + (v· V)u), V v E D(A),
where II is the projector in L2(f2)l onto the space H.
Theorem 2.1 applies to the prepared form of this equation (i.e. in the form of
Theorem 3.1). The inertial manifold is obtained as a graph above a rootspace of A,
namely the space spanned by the root vectors VI, •.. , Vn of A, with n sufficiently large.
Also, as indicated before, and in view of (4.8), the inertial manifold M contains 0 in
D(A"') and is tangent at 0 to PE.
As indicated before, a similar concept appears, in the meteorology literature, for
the equations of meteorology, under the name of slow manifold.
REFERENCES
S. Agmon (1965), Lectures on Elliptic Boundary Value Problems, Mathematical Studies,
Van Nostrand, New-York.
M.S. Agranovitch (1977), Series in the root vectors of operators that are very close to being selfadjoint, Thnct. Anal. and its Appl., 4, 296-299, Translated from Funktsional'nyi
Analizi Eg 0 Prilozheniga, 11, 1977,65-67.
S.N. Chow and K. Lu (1988), Invariant manifolds for flows in Banach spaces, J. Diff.
Equ., 74, 285-317
P. Constantin, C. Foias, B. Nicolaenko and R. Temam (1988), Integral Manifolds and
Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, NewYork, Applied Mathematical Sciences Series, Vol. 70.
R. Courant and D. Hilbert (1953), Methods of Mathematical Physics, Intersciences
Publishers, New-York.
(and (4.31) is satisfied). The existence of u,p, follows e.g. from J.L. Lions (1969), R.
Temam (1977). Let u be any such solution and set v = u - u and q = p - p. We have
av at + e( -~tv - v~v + (u. V)v + (v· V)u + (v· V)v + Vq = 0,
V·v=O,
v(x+Le;,t)=v(x,t), q(x,Le;,t)=q(x,t), i=I, ... ,f.
(4.36)
(4.37)
(4.38)
Equations (4.36)-(4.38) amount to an equation for v of type (4.4). Here we have
D(A) = D(Ao) and
Av = e( -~tv + II«u· V)v + (v· V)u), V v E D(A),
where II is the projector in L2(f2)l onto the space H.
Theorem 2.1 applies to the prepared form of this equation (i.e. in the form of
Theorem 3.1). The inertial manifold is obtained as a graph above a rootspace of A,
namely the space spanned by the root vectors VI, •.. , Vn of A, with n sufficiently large.
Also, as indicated before, and in view of (4.8), the inertial manifold M contains 0 in
D(A"') and is tangent at 0 to PE.
As indicated before, a similar concept appears, in the meteorology literature, for
the equations of meteorology, under the name of slow manifold.
REFERENCES
S. Agmon (1965), Lectures on Elliptic Boundary Value Problems, Mathematical Studies,
Van Nostrand, New-York.
M.S. Agranovitch (1977), Series in the root vectors of operators that are very close to being selfadjoint, Thnct. Anal. and its Appl., 4, 296-299, Translated from Funktsional'nyi
Analizi Eg 0 Prilozheniga, 11, 1977,65-67.
S.N. Chow and K. Lu (1988), Invariant manifolds for flows in Banach spaces, J. Diff.
Equ., 74, 285-317
P. Constantin, C. Foias, B. Nicolaenko and R. Temam (1988), Integral Manifolds and
Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, NewYork, Applied Mathematical Sciences Series, Vol. 70.
R. Courant and D. Hilbert (1953), Methods of Mathematical Physics, Intersciences
Publishers, New-York.