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Inertial Manifolds
a)
A F ( p , t ) £ c1 N -1 / 2 ;
A ( F ( p1 , t ) - F ( p2 , t ) ) £ c2 A ( p1 - p2 )
for any p , p1 , p2 Î P D ( A) ;
b)
for any solution u ( t ) Î D ( A ) to problem (6.21) there
exists t * > 1 such that
3
AQ ( u ( t ) - F ( Pu ( t ) , t ) )
£
£ c 3 exp { -s 0 N 1 / 2 ( t - t* ) } + c 4 exp { -s 1 N 1 / 2 } .
Here P is the orthoprojector onto the first N eigenelements of the
operator A .
E x e r c i s e 6.5 Use Theorem 6.1 to construct approximate inertial manifolds
for (a) the nonlocal Burgers equation, (b) the Cahn-Hilliard equation, and (c) the system of reaction-diffusion equations (see Sections 3 and 4 of Chapter 2).
In conclusion of the section we note (see [8], [9]) that in the autonomous case the approximate IM can also be built using the equation
á F¢( p ) ; - Ap + P B ( p + F ( p ) )ñ + A F ( p ) = Q B ( p + F ( p ) ) .
(6.22)
Here p Î PH , Q = I - P , F¢ ( p ) is the Frechét derivative and á F¢( p ) , wñ is its
value at the point p on the element w . At least formally, equation (6.22) can be obtained if we substitute the pair { p ( t ) ; F( p ( t ) ) } into equation (6.3). The second of
-1
equations (6.3) implicitly contains a small parameter lN
+ 1 . Therefore, using (6.22)
we can suggest an iteration process of calculation of the sequence { Fm } giving the
approximate IM:
A Fk ( p ) = Q B ( p + Fn ( k ) ( p ) ) +
1
+ á Fn¢
2 (k)
( p ) ; Ap - P B ( p + Fn
3 (k)
( p ) )ñ ,
k ³ 1,
(6.23)
where the integers ni ( k ) are such that
0 £ ni ( k ) £ k - 1 ,
lim n i ( k ) = ¥ ,
k®¥
i = 1, 2, 3 .
One should also choose the zeroth approximation and concretely define the form of
the values ni ( k ) (for example, we can take F0 ( v ) º 0 and n i ( k ) = k - 1 , i = 1 , 2 ,
3 ). When constructing a sequence of approximate IMs one has to solve only a linear
stationary problem on each step. From the point of view of concrete calculations this
gives certain advantages in comparison with the construction used in Theorem 6.1.
However, these manifolds have the power order of approximation only (for detailed
discussion of this construction and for proofs see [9]).
E x e r c i s e 6.6 Prove that the mapping F1 ( v ) has the form (6.4) under the
condition F0 ( v ) º 0 . Write down the equation for F2 ( v ) when
n1( 2 ) = 1 , n2 ( 2 ) = n 3 ( 2 ) = 0 .