On the longtime behavior of
solutions to a model for
epitaxial growth
Atsushi Yagi
Hideaki Fujimura
Jian Luca Mola
Maurizio Grasselli
Model Equation
We are concernedwith a forth- order equation
 u 
u
2
  a u    
,
2 
t
 1 | u | 
  (0, )
in a two - dimensional bounded domain  ( R 2 ) .
a  0 and   0 are (positive)constants.
Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr,
Phys. Rev. Lett. (1994)
Surface Diffusion
u
O
r
u
O
r
u
 a2u
t
Simplification due to W.W. Mullins (1957)
Schwoebel Effect
u
O
r
u
O
r
 u 
u

    
2

t
1


u


Negative diffusion due to M.D.Johnson et al. (1994)
Epitaxial Growth Model
Our problem is:
 u
 u 
2

   a u    
2 
 1 | u | 
 t
 u 
(EGM)   u  0
 n n

 u( x,0)  u0 ( x )

in   (0, ),
on   (0,),
in ,
in two - dimensional bounded domain  ( R 2 ) of C 4 class.
Analytical Results
1. Fundamentals
2. Convergence as t  
3. Finite dimensional Attractors
4. Homogeneous Stationary Solutions
Abstract Formulation
Abstract parabolic semi-linear evolution equation:
 du
  Au  F (u ),
(E)  dt
 u (0)  u0 ,
0  t  ,
in an underlying space X  L2 (), where
7/2

D
(
F
)

H
(),
 D( A)  {u  H 4 (); u   u  0 on },

n n
 u 
and 


.
 Au  a2u ,
 F (u )     
2
1

|

u
|



Global Solutions and Dynamical System
Global existence:
u0  H 1 ()  ! u global sol. to (E) such that
u  C ((0, ); H 4 ())  C ([0, ); H 1 ())  C1 ((0, ); L2 ()).
For 0  t  , we put S (t )u0  u (t ; u0 ), u0  H 1 (). Then,
S (t ) S ( s )  S (t  s ), S (0)  I , " nonlinear semigroup on H 1 ()".
So , ( S (t ), K , L2 ()) generates " a dynamical system"determined from
(E), where K  H 1 () denotes the phase space.
Lyapunov Function
Put

a | u |2   log (1 | u |2 ) dx,

 (u )  
2
uHN
().
Then, along the trajector y we see that
d  ( S (t )u )  2
u 2 dx  0, i.e.,  ( S (t )u )  as t  .
0
0
dt
 t

In addition, " u  H 4 2 () is a stationary solution t o (E)"   ' (u )  0.
N
Therefore, (u) becomes a Lyapunov function
S(t)u0  ū as t  
Simon-Łojasiewicz
theory:
For u0  H 1 (), let
u  ω(u0 ) 
 {S ( s)u ; t  s  }, i.e.,  (u )  0.
0
0t 
Assume it holds true that
(u )  (u )
L2
   (u )   (u )
1
,
u  B L2 (u ; r ),
with some 0    1 / 2,   0, r  0. T hen, as t  ,
S (t )u0  u . " stationarysolutionof (E)"
Finite Dimensional Attractors
Eden, Foias, Nicolaenko,Temam (1994) introduced:
M is called " an exponentia l attractor" of ( S (t ), K , L2 ()) if :
 M is a compact set of L2 () with finite fractal dimension;
 M contains the global attractor A;
 S (t ) M  M for every 0  t  ;
    0,  B : bounded set of K ,  C B  0
h( S (t ) B, M )  C B e  t ,
where h ( B1 , B2 )  sup
0  t  ,
inf
uB1 vB2
uv.
h
B1
B2
Fractal Dimension
Let M be a comact set of a Banach space X .
N ( ) : a minimal number of  - balls of X necessary for covering M ,
i.e.,
M
N ( )

n 1
B( xn ;  ),
xn  X .
Then,
d F ( M )  lim
 0
log N ( )
log 
1
.
Mañé-Hölder type embedding:
If d F ( M )  , then M  C N with a Holder continuous isomorphis m.
Exponential attractors for (E)
L2, m ()  { f  L2 (); m( f )  |  |1  f dx  0},

H 1m ()  {u  H 1 (); m(u )  |  |1  u dx  0},

Note that m(u0 )  0  m( S (t )u0 )  0 for every 0  t  .
(E) generates a dynamical system
( S (t ), H 1m (), L2, m ()) .
(S(t),Hm1(Ω),L2,m(Ω)) has exponential attractors.
Stationary Solutions
Linearized principle:
( S (t ), Z , Z ) : dynamical system in a Banach space Z
u : an equilibrium of ( S (t ), Z , Z )
Assume :  t   0, S (t  )  C 1, (W , Z ) (0    1) in a neighborhood W
of u with
 ( S (t  )' u )  {  C; |  |  1}   . " spectralseparation"
T hen,
  ( S (t  )' u )  {|  |  1}  u is " stable".
  ( S (t  )' u )  {|  |  1}    u is " unstable".
Stability-Instability of 0
Let 0  1  2  3   be the eigenvalue s of an operator
   " Neumann boundary conds. on ".
 If   a1, then the 0 is stable.
 If N  #{k ;   ak }  0, then the 0 is unstable.
We have the estimate dF(M)  N.
Summary
• (EGM) generates a dynamical system
(S(t),H1(),L2()).
•  u0 H1(), S(t)u0  ū as t  ,
where ū is a stationary solution of (EGM).
• The set {stationary solutions}  M (finite
dimensional attractor).
• If  < a1, the homogeneous stationary
solutions are stable; otherwise, unstable.
Problems
• How is the structure of
stationary solutions?
• What is a profile of stationary
solution?
Thank You for Your Kind Attention

STAT435 - Lecture 6 - University of Otago

Montréal, QC les 27–28 mai, 2015