I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Polynomial decay rate
for the Maxwell’s equations with Ohm’s law
Kim Dang PHUNG
Sichuan University.
November 2010, Paris, IHP
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Maxwell’s equation with Ohm’s law
Let
be a smooth bounded domain in R3 .
8
"o @t E curlH + (x) E = 0
>
>
<
o @t H + curlE = 0
div ( o H) = 0
>
>
:
E
j@ = H
j@ = 0
2 L1 ( )
take "o =
o
=1
and
0
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Energy
E (t) =
E (t2 )
1
2
Z
E (t1 ) =
E1 (t) =
1
2
Z
jE (x; t)j2 + jH (x; t)j2 dx
Z
t2
t1
Z
(x) jE (x; t)j2 dxdt
j@t E (x; t)j2 + j@t H (x; t)j2 dx
0
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Di¢ culties
Free divergence is not preserved by the system.
@t2 E + curl curl E + @t E = 0
curl curl E =
E+r div E
and
div E 6= 0 on
but
div E = 0 on
f (x) = 0g
from now
div E ( ; t = 0) = 0 on
f (x) = 0g
(0; +1)
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Remedies
"scalar potential, vector potential and Coulomb gauge". Suppose
is simply connected and @ has only one connected component.
E = rp @t A
H = curl A
and
8 2
< @t A + curl curl A = @t rp + E
div A = 0
:
A
=0
j@
kEk2L2 (
)3
= krpk2L2 (
k@t rpkL2 (
)3
)3
+ k@t Ak2L2 (
k EkL2 (
)3
)3
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Results
(x) constant > 0
(x) = 0
8x 2 !
8x 2 n!
=) treatment of the divergence part.
KNOWN RESULTS
lim E (t) = 0
t!+1
" GCC " = no trapped ray
=)
E (t)
ce
t
E (0)
NEW RESULT
! = small neighborhood outside parallel trapped ray
C
parallel trapped rays =) E (t)
(E (0) + E1 (0))
t
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Interpolation observation inequality
Polynomial decay
E (0)
=)
8h > 0
Z ( C )1= Z
h
0
jEj2 dxdt + h (E (0) + E1 (0))
Now,
(=
also true
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Interpolation observation inequality for the Wave equation
8 2
u=0
< @t u
uj@ = 0 or @ uj@ = 0
: 1
2
E (u; 0) = k(r; @t ) u ( ; 0)k2L2 (
c ku ( ; t)kL2 ( )
)4
! = small neighborhood outside parallel trapped ray
For parallel trapped ray,
E (u; 0)
Z ( C )1= Z
h
0
!
8h > 0
j(u; @t u)j2 dxdt + h (E (u; 0) + E (@t u; 0))
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
New operator
h 2 (0; 1], L
1. Add new variable s .
i@s + h
@t2
F =0
F (x; t; 0) = " u (x; t) "
Multiply by u over (x; t; s) 2
parts,
[ T; T ]
term at s = 0
= term at s = L
+ h term at t = T
+ h term at @ \ (outside parallel @ )
+ h term at @ \ (parallel @ )
[0; L] and integrate by
! closed to u (x; t)
! term on !
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
New operator
h 2 (0; 1], L
1. Add new variable s .
i@s + h
@t2
F =0
F (x; t; 0) = " u (x; t) "
Multiply by u over (x; t; s) 2
parts,
[ T; T ]
term at s = 0
= term at s = L
+ h term at t = T
+ h term at @ \ (outside parallel @ )
+ h term at @ \ (parallel @ )
[0; L] and integrate by
! closed to u (x; t)
! term on !
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Fourier transform
Fourier inversion formula
f (x; t) =
1
(2 )4
Z
ei(x
+t )
R4
fb( ; ) d d
Fourier Integral Operator
F (x; t; s) =
1
(2 )4
Z
R4
e i(j
j2
2
)hs a (x
xo
2 hs; t + 2 hs; s)
ei(x
+t ) fb(
with a (x; t; s) smooth and localized around (x; t) = (0; 0).
; )d d
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Properties
When s = 0 ,
F (x; t; 0) = a (x
If i@s + h
@t2
xo ; t; 0) f (x; t)
a=0,
i@s + h
@t2
F =0
Now, it remains to take a good a (x; t; s) solution of
i@s + h
@t2 a = 0 to treat
the term at s = L
the term at t = T
the term at @ \ (parallel @ )
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Construction of a = a (x; t; s) 2 C 1 R5 ; C
0
10
jxj2
B
C
is + 1 C B
B e
e
B
CB
B
a (x; t; s) = B
C
B
B
C@
(is + 1)3=2
@
A
1
4h
1
4
p
1
t2
ihs + 1 C
C
C
C
ihs + 1
A
Then
i@s + h
@t2
e
h s2 + 1
a (x; t; s) = 0
t2 =4
jxj2 =4
ja (x; t; s)j =
p
s2 + 1
3=2
(hs)2 + 1
e
q
1=2
(hs)2 + 1
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
At the end
h 2 (0; 1], L 1. Add new variable s .Construction of
a = a (x; t; s) 2 C 1 R5 ; C .
i@s + h
@t2
F =0
F (x; t; 0) = " u (x; t) "
Multiply by u over (x; t; s) 2
parts,
[ T; T ]
term at s = 0
= term at s = L
+ h term at t = T
+ h term at @ \ (outside parallel @ )
+ h term at @ \ (parallel @ )
[0; L] and integrate by
!
!
!
!
!
closed to u (x; t)
small for large L 1
small for large T 1
term on !
by microlocal analysis
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
At the end
h 2 (0; 1], L 1. Add new variable s .Construction of
a = a (x; t; s) 2 C 1 R5 ; C .
i@s + h
@t2
F =0
F (x; t; 0) = " u (x; t) "
Multiply by u over (x; t; s) 2
parts,
[ T; T ]
term at s = 0
= term at s = L
+ h term at t = T
+ h term at @ \ (outside parallel @ )
+ h term at @ \ (parallel @ )
[0; L] and integrate by
!
!
!
!
!
closed to u (x; t)
small for large L 1
small for large T 1
term on !
by microlocal analysis
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
Localization in Fourier variables
Partition for
3
2(
F (x; t; s) =
e i(j
j2
2
)hs a (x
1;
o3
1
(2 )4
xo
o3
+ 1),
Z Z
R2
o3 +1
o3
1
o3
2 (2Z + 1) and j j < ,
Z
j j<
2 hs; t + 2 hs; s) ei(x
+t ) 'u
c(
; )d d
with a (x; t; s) smooth and localized around (x; t) = (0; 0), ' (x; t)
smooth.
I: Maxwell’s equations with Ohm’s law.
II: Polynomial energy decay.
References
F. Cardoso and G. Vodev, On the stabilization of the wave
equation by the boundary, Serdica Math. J. 28 (2002), 233-240.
N. Burq and M. Hitrik, Energy decay for damped wave equations
on partially rectangular domains, Math. Res. Lett. 14 (2007),
35-47.
H. Nishiyama, Polynomial decay rate for damped wave equations
on partially rectangular domains, Math. Res. Lett. 16 (2009),
881-894.
K.-D. Phung, Polynomial decay rate for the dissipative wave
equation, J. Di¤erential Equations 240 (2007), 92-124.
K.-D. Phung, Boundary stabilization for the wave equation in a
bounded cylindrical domain, Discrete Contin. Dyn. Syst. 20
(2008), 1057-1093.
K. D. Phung, Contrôle et stabilisation d’ondes électromagnétiques,
ESAIM Control Optim. Calc. Var. 5 (2000), 87-137.
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