llog l - université sorbonne paris nordvauchelet/ddsplog.pdf!#+j"%+ u ')@n4h!#+ @h6
TRANSCRIPT
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ρk(t, x) =Ns(t, x)
Z(t, x) e−�k(t,x) = u(t, x)e−�k(t,x),
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V ∞ ∈ C2(Ω) &s@%+J+�, § | 7y, 0C+¶c
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L2+J@D45'AFT2�4H+T6,(Ns
c�¤a( :s2��-4J7vF 0#X45')S#X)CE'A(U*45!#+T"H':;4% D∇Ns dx +1
2
∫
ω
(∇(N2s ))> D∇Vs dx = 0.
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&s'A(B� +J�¶45')6,( cµzT6�:7,/.40Z+�0#@%+ @%4H"H6,(U*,X)Cf4H!#+ :s2��¶445!32�4FT6�@%4 6�:K45!#+4H+J"HFW@Z�!U'A��!p2�S#Sy+�2�" 'A(^4H!#++]\USU"H+J@%@H')6,(p6�:K45!#')@ /1032�(�45'4aC2�"H+(#6�(#(#+J*,2�<45'[,+ +]c�©45!#+-"%�')@H+�73Z+ !#2[�+ 4H6T+J2�XK�')4H!¢4H!#+ /1032�(14H')4aC −div x(D∇xVs) c`!U+J"H+¶:L6,"H+�7y4H!#+@HS32��J++-S?+-(#+J(U�J+�6�:
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L log LO?6,"%(¡6,(¡45!#+
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`!U+J(p45!#+ "H+-*,0#XA2�"H')_J+J&p@DC @D45+-F(DDSPε)
')@+ 23(U+J&�:L6�"ε ∈ [0, 1] O�C 0,
N εs (0, x) = Nε,0s (x) := min{N0s , ε−1}, x ∈ ω
∂νNs(t, x) = 06,(
∂ω × (0, 1),
(SPε)
−12
∂2zχεk + R
ε[V ε]χεk = �εkχεk (k ≥ 1),
χεk(t, x, ·) ∈ H10 (0, 1),∫ 1
0
χεk χε` dz = δk` ,
−∆x,zV ε = Rε[∑
k
N εse−�
εk
Zε |χεk|2]
,
∂νV (t, x, z) = 06,(
∂ω × (0, 1), V (t, x, 0) = V (t, x, 1) = 0 :L6," x ∈ ω.��Er#�w P8!P��������
ε = 0 J I �B�D"989� R0 = Id "9�O'��
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d
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(ρk log ρk − ρk) dx +1
2
d
dt
∫∫
Ω
|∇x,zV |2 dxdz
+d
dt
∑
k
∫
ω
ρk(�k − 〈|χk|2Rε[V ]〉) dx = −∫
ω
(∑
k
e−�k
)(∇xu)>D∇xu
udx.
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�k − 〈|χk|2Rε[V ]〉 =1
2〈|∂zχk|2〉.
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�w� ��y��t;nwts�yx �K�
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Ns ∈ L∞(0, T ; L1(ω)) U"9���! �*�εN
�w�8��5 `!#')@+¶\ 'A@D45+-(#�J+W"H+-@H0#X4 'A@6,OU452�'A(U+J&XA')�+²')( ,/.10v45!32�(UE@4H6�2fFW'A(#')FT')_�2�4H'A6,(p6�:2�J6,(�[�+¶\¢:L0#(#�¶45'A6�(32�XYcv`!#'A@:L0#(#�¶45')6,(32�X?'A@+;23(#+J&�6,(
H101O�C
J(V ) =1
2
∫∫
Ω
|∇V |2 dxdz +∫
ω
Ns log∑
k
e−�k[Rε[V ]] dx = J0(V ) + J1(V, Ns).
&szUc § ?+
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L log L+-@%45')F^2�45+6,(i45!U+I@%0#"%:s2��J+¬+J(U@H')4aC�7Z+
(#+J+-&�4H6T0#@%+45!#+ `�"H0#')(#*,+-"')(#+J/1032�XA')4aC& § c ?+2�(U&¢@%!32�"%S?+-"+-@%4H'AFT2�45+-@6�(p4H!#+ +J')*,+J(�[�2�XA0U+J@6�:v4H!#+.� ��!#"%$ &U'A(#*,+-"�6,S?+-"52�4H6," �k[V ]
@D4�2�4H+J&¢'A( &sM c § +]c `!#+J(
|J1(V, Ns) − J1(Ṽ , Ns)| ≤∫
ω
Ns(x) supk
(|�k[Rε[V ]] − �k[Rε[Ṽ ]]|(x)) dx
≤ C∫
ω
Ns(1 + ‖Rε[V ]‖1/2L2z(0,1) + ‖Rε[Ṽ ]‖1/2L2z(0,1))‖R
ε[V − Ṽ ]‖L2z(0,1) dx.&szUc § >+¤a(45!#+ :L6,X)XA6�'A(U*
C2�(#&
CT�')XAX�@%452�(#&:L6,"(#6�(#(#+J*,2�45'[,+ �-6,(#@%452�(�45@ +-S?+-(#'A(U*¢6�(#X)C¢6,(
45!#+ &32�4�22�(#&p(#6�4�6�(εc�= +!32�[,+.@H+-+J(¢'A( & § c >+V4H!32�44H!#+ `"%0#'A(U*,+J"')(#+J/1032�X)')4aC^2�(#&��
'AFWS#X)C
∫
ω
Ns‖Rε[V − Ṽ ]‖L2z(0,1) dx ≤ CT‖Rε[V − Ṽ ]‖H1(Ω) ≤ CT‖V − Ṽ ‖H1(Ω),&szUc § .?+
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45!32�(#1@456W45!#+ S#"H6,Sy+J"D45')+J@6�:R45!U+�"H+-*,0#XA2�"H')_�2�4H'A6,(Rε&´PK+JFWFT2^ c § +]c�(N6,'A(#*W45!#+ @52�FW+ :L6,"
45!#+ 6�4H!#+J"%@4H+J"%F�7UZ+ !32�[,+∫
ω
Ns‖Rε[V ]‖1/2L2z(0,1)‖Rε[V − Ṽ ]‖L2z(0,1) dx ≤ C‖V ‖
1/2H1(Ω)‖V − Ṽ ‖H1(Ω)×
∫
ω
(Ns log Ns − Ns + exp
γ‖Rε[V ]‖1/2L2z(0,1)‖Rε[V − Ṽ ]‖L2z(0,1)
‖Rε[V ]‖1/2H1(Ω)‖Rε[V − Ṽ ]‖H1(Ω)
)dx,
�!#+J"%+γ'A@^4H!#+B�J6�(#@%452�(�4f+ 2#(#+J& 'A( 4H!#+B`�"H0#&U'A(#*,+-"'A(U+J/1032�X)')4aC & § c ?+]c MS#SUX)CE'A(#* 45!#+
£ 2�0#��!�C�
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�w� ��y��t;nwts�yx �KQP �]o �?xynwt´x j t;n]� ��� ���ε ∈ [0, 1] "9�O' T > 0 N3XM $ ��&%�� Ns
"9�O'Ñs
"94K�L9�.89��� �.�
L∞(0, T ; L1(ω))"9�O') U"9���!
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ZCf45!U+ �N$,X)+J"')(#+J/1032�X)')4aC�7
∫
ω
|Ns − Ñs|(1 + ‖Rε[V ε]‖1/2L2z )‖Rε[V ε − Ṽ ε]‖L2z dxdz ≤
≤ ‖Ns − Ñs‖1/4L1(ω)(∫
ω
(Ns + Ñs)(1 + ‖Rε[V ε]‖2/3L2z )‖Rε[V ε − Ṽ ε]‖4/3L2z dx
)3/4.
&szUc �~ +`!#+ 2�@%@H0#FWSU45')6,( �-²'AFWS#XA')+J@V2�Oy6,0#(#&')(
L log L6�:K4H!#++-(#@H'45')+J@
Ns2�(#&
Ñsc`!10#@Z+
��2�(I:L6,XAX)6 4H!#+'A+J2f6�:4H!#+S#"H6E6�:Z6�:G"H6�S?6,@%')4H'A6,(¬zUc § 2�(#&IS#"H6[�+^2�:;4H+J" @%4H"52�')*,!�4H:L6,"D 2�"H&��2�X)�J0#XA2�45')6,(#@�0#@H')(#*4H!#+ZbV6,0#(U*'A(#+-/1032�XA'4aC & § c |>+�2�(#&�4H!#+ `�"H0#')(#*,+-"v'A(U+J/1032�X)')4aC & § c ?+]7�45!#2�4∫
ω
(Ns+Ñs)(1+‖Rε[V ε]‖2/3L2z )‖Rε[V ε−Ṽ ε]‖4/3L2z dx ≤ C2(1+‖V
ε‖2/3H1(Ω))‖V ε−Ṽ ε‖4/3
H1(Ω),
�!#'A��!W'A@Oy6,0#(U+J&W@H')(#�J+V ε
2�(#&Ṽ ε
2�"%+O?6�0#(#+-&T')(H1(Ω)
�'45!WG"H6,Sy6,@%')45')6,(zUc § cv`!10#@XA6E6,1'A(#*^2�4B&szUc �z?+¶7Z+ !32�[,+²S#"%6[,+J&4H!32�44H!#+J"%+ +¶\ ')@%45@2^(#6,(#(U+J*�2�4H')[�+��J6�(#@%452�(�4
C3@H0#��!
45!32�4‖V ε − Ṽ ε‖2H1(Ω) ≤ C3‖Ns − Ñs‖1/4L1(ω).
`!#'A@�X)+�2�@4H6 &szUc § +¶c36," 45!U+N�-6,(�[,+J"%*,+J(U�J+.2�@
ε → 0 7EZ+�-6,(#@H')+J" V ε 2�(#& V 45!#+N@H6,X)0U45')6,(#@ 6�:84H!#+0U(#"H+-*,0
-
2�(#&|II| ≤ C2(1 + ‖V ε‖H1(Ω) + ‖V ‖H1(Ω))‖(Rε − Id)[V ε]‖H1(Ω),
3"H6,F 45!#+S#"%6,Sy+J"%4H'A+-@6�: 45!#+"%+J*,0UX2�"%'A_�2�45'A6�(Rε&L@H+J+¬PK+-FTFT2¡ c § +¶7�Z+I&U+J#�-+¬+�2�@%'AX)C
45!32�4�7R@%'A(#�-+V ε
'[email protected],0#(#+-& 'A(H1(Ω)
')(#+JSy+J(U+J(�45XCI6�:ε7Z+!32�[,+
limε→0 |I| = 0 =limε→0 |II|
c�v')(32�XAXC,72�@ Z+!32�[�+f@H!#6�( :L6,".45!#+T@H+J�-6,(#&¡45+-"HF 6�:4H!#+"H')*,!�4�!#2�(#& @%'A+6�:7&szUc ,>+¶7E45!#+4H+J"HF
III'A@�(#6,(E+')FTS#X)'A+-@
45!32�4limε→0
‖V ε − V ‖L∞(0,T ;H1(Ω)) = 0.
� ��� �!�� �*��,(���������"-�1� ���A���
� ��� ���7��� ��
�� ���-����%*� �������$� � � � ���� �� �=�$%�� ����
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T > 0"9�O'
ε ∈ (0, 1) +�� ;7@&�('VN ������ �.���?4$��L9��H "94������(')6O4K�9+UH!��%(DDSPε) I
�.�.���+�� & § cµz?+0"V'9%3�.�. "3�&�����U��� 1��H �&�������(N εs , V
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0#(#1(#6�(N̂s5
∂tN̂s − div x(D(∇xN̂s + N̂s∇xVs)) = 0,�')4H!²45!#+�'A(U')45'A2�X1�J6,(U')4H'A6,(
N0s2�(U&4H!#+�O?6�0#(#&32�"DC²�J6�(#')4H'A6,(
∂νNs = 0:L6�"
x ∈ ∂ω c
=+�J2�( SU"H6[,+W2�@')( , .10R4H!32�4F
')@ 2f�J6,(�45"H2��-4H'A6,(6�(I4H!#+²@H+¶4 ST0:L6,"
T0@%F^2�XAXv+-(#6,0#*�!8c
`!10#@NZ+ !32�[,+ 2@%6,XA0 45'A6�(f6,([0, T0]
4H!32�4�Z+.�J2�(¢+¶\E45+-(#&�4H6[0, T ]
c
§ z
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� ��� � ������� � � ��� ���$ %�� �!� �ε → 0
=+!#2[�+(#6 2�XAXyF^2�4%45+-"H@4H6 �J6,(U@%45"%0#�-4�@%6,XA0 45'A6�(#@6�:94H!#+0#(#"%+J*,0UX2�"%'A_J+-&fS#"H6�O#XA+-F 2�(U&456S#"H6[�+ `!#+-6,"H+-F § cµzUc�= +�'AXAXy0#@%+ 4H!#+M0UO#'A(
-
=+ 2�(�4 (#6 456^S32�@H@N456^4H!#+ XA')FT'4ε → 0 'A(¢45!#+�"H':;4%?0#@H')6,(+J/1032�4H'A6,( 2�@
ε → 0 cf3"H6,F P9+-FTFT2M�c § 7#Z+ !32�[�+∇x�εk[V ε] = 〈|χεk|2∇xV ε〉.`!#+ �J6,(�[�+J"H*�+J(#�-+ 6�:
V ε'A(
L2(0, T ; H1(Ω))2�(#&4H!#+XA6E��2�X?PK'AS#@%��!#')4H_+-S?+-(#+J(U�J+.6�:945!#+
+J')*,+J(�[,+-�-4H6,"H@T6�:N4H!#+ �2�FW'AX456,(U'2�(I�')45! "H+-@HSy+J�¶4T4H6¬4H!#+¢S?6�4H+J(�45'A2�X &L@H+J+�PK+JFWF^2BM cµ~I6�:MS#Sy+J(U'\¬6�:$, .10C+2�XAX)6 0#@456f�J6�(#�JX)0#+ 45!#2�4 ∇x�εk[V ε] → ∇x�k[V ]
'A(Lp((0, T ) × ω):L6,".2�XAX
p ∈ [1, 2) cT3"%6,F £6,"%6,XAXA2�"%C�M�cµzU79+W+J&U0#�J+45!32�4.:L6�" 2�XAX q ∈ [1, +∞) 7KZ+T�J2�(+¶\E45"H2��-42^@H0#OU@H+J/10#+-(#�J+W@H0U��!B45!32�4 �εk[V ε] → �k[V ]'A(
L2(0, T ; Lq(ω))c`!E0U@J7K@H')(#�J+.+
!32�[,+|e−�εk [V ε] − e−�k[V ]| ≤ e− 12π2k2|�εk[V ε] − �k[V ]|,+.+-#�J+ 45!#2�4:L6,"�2�(�C
r ∈ [1, p)
∇xV εs =∑
k ∇x�εke−�εk
Zε�J6,(�[�+J"H*�+J@')(
Lr((0, T ) × ω) c`!#'A@')@+J(#6�0#*,!p456WS#"H6[,+
N εs∇xV εs ⇀ Ns∇xVs'A( D′([0, T ] × ω).
`!10#@ 0#S^4H6�2�(^+¶\E4H"52��¶45'A6�(87(Ns, V )
'A@Z2 @%6,XA0 45'A6�(W6�:(DDSP )
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-
M(#&p:L6,"�2Uc +�ct ∈ [0, T ] 7
W (t) +
∫ t
0
D(s) ds ≤ lim infε→0
(W ε(t) +
∫ t
0
Dε(s) ds).
� � �A� � � � � ����� � �*���A�1�
`!#'A@²@%+J�¶45'A6�( ')@²+-[�6�45+-& 4H645!#+fS#"%6 6�:N6�:N`!U+J6,"%+JF § c ~#c6=+f0#@%+¢2�(�+J(�45"%6,S�C¡FT+¶45!#6E&"H+-X)CE'A(#* 6,( 45!#+ PK6,*�2�"%')45!UFT')�� 6,Oy6,X)+-[ 'A(U+J/1032�X)')45')+J@f2�(#&�6,( 45!#+£@H')@H_J±�"D+
=+."H+-��2�X)X 45!U+��"H6,@%@�XA6,*�2�"H')4H!#FW'A�� 6,Oy6,XA+¶['A(U+J/1032�X)')4aC
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�6,"%+J6[,+-"J7345!#+.�EXA6�4HO?6E6,F [�2�"H'A2�O#XA+:L6,"4H!#+ @%4�2�45'A6�(32�"%CTS#"%6,O#XA+-Fu∞
'A@ 2�-6,(#@D4�2�(�4�c `!10#@
u∞
u=
(u∞
NI
∫
ω
Z(x) dx)
=∑
k
∫
ω
u∞e−�kdx
NI`!#+J"%+-:L6,"%+.�'45!f45!U+.')+J(�4H')4aC
u∞ = ρ∞k e−�∞
k73O�Cp2W®�+J(#@%+J(B')(#+J/1032�X)')4aC�7 Z+ !32�[�+
logu∞
u= log
∑
k
∫
ω
ρ∞k e�∞
k−�k
dx
NI≥∑
k
∫
ω
(�∞k − �k)ρ∞kNI
dx.
& = + "H+J�J2�XAX?45!32�4 ∑k
∫ω
ρ∞k = NI+¶c 'A(32�X)X)CZ+ !32�[�+ 6,O 4�2�')(#+J&
d
dtW∞(t) ≤ −αλ(
∑
k
∫
ω
ρk logρkρ∞k
dx +∑
k
∫
ω
(ρk − ρ∞k )(�k − �∞k ) dx)
≤ −αλ(∑
k
∫
ω
ρk logρkρ∞k
dx +
∫∫
Ω
(N − N∞)(V − V ∞) dxdz
+∑
k
∫
ω
ρk(�k − �∞k − 〈|χk|2(V − V ∞)〉) dx
+∑
k
∫
ω
ρ∞k (�∞k − �k − 〈|χ∞k |2(V ∞ − V )〉 dx).
`!#+BXA2�@%4W45+-"HF 6�:4H!#+¥"%'A*,!�4T!32�(#& @H'A&U+¢�!#')��!¦'A@W@H')FT')X2�" 4564H!#+�XA2�@%4W45+-"HF 6�:W∞
')@(#6,(#(U+J*�2�4H')[�+�c�M�J�-6,"H')(#*456T4H!#+ G�6,'A@%@H6,(�+J/1032�4H'A6,( 74H!#+.@H+J�-6,(#&B4H+J"HF 6�:R45!#+."H')*,!�4!32�(#&@H')+ 'A@+J/1032�X8456
∫∫|∇x,z(V − V ∞)|2 dxdz.
`!10#@J7+ !32�[�+ @%!#6�(�45!#2�4
d
dtW∞(t) ≤ −αλW∞(t),
2�(#& 2 �"H6,(�2�XAX 4aCESy+B2�"%*,0#FW+J(�4TCE')+JXA&U@4H!#+¥+]\ S?6,(U+J(�45'A2�X��J6,(�[,+-"H*,+-(#�J+¥6�:45!U+�"%+JX2�45')[�++J(�45"%6,S�C35
W∞(t) ≤ W∞(0) e−αλt c +¶\E4�+@%!32�X)X80#@H+4H!#+ £@H')@H_�±�"D
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� � � ����� �*�� � � ��� �*���� � � � ����*,.���� � ��� � � ��� �A� ��"�� � ���!���
`!#+N2�'AF 6�:?4H!#'A@MNS#Sy+J(#'\'A@R4H6 S#"H+-@H+J(�4Z@H!32�"HSW+J@%4H'AFT2�45+-@ 6,(²45!#+�+J')*,+J(�[�2�X)0#+J@V2�(#&+-'A*,+-( <:L0#(#�¶45'A6�(#@6�:4H!#+ 6,(#+]
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�!#+J"%+Ak
')@ 2 (#6�(#(#+J*,2�45'[,+�-6,(#@D4�2�(�4456�O?+&U+-45+-"HFW'A(#+-&8c`!32�(UE@ 4H6²2£ 2�0#��!�C�+'A(f45!U+.')+J(�4H')4aC ∫ 1
0χ2k dz = 1
7UZ+ 6�OU4�2�')(
1 ≥ A2k∫ 1
0
sin(√
2�kz)2 dz + 4Ak∫ 1
0
sin(√
2�kz)∫ z
0
sin(√
2�k(z − t))√2�k
V (t)χk(t) dtdz.&LM�c |>+
36," 45!#+ @%+J�J6�(#&�4H+J"HF +.!32�[,+:L"H6�F &sM c z?+∣∣∣∣∫ 1
0
sin(√
2�kz)∫ z
0
sin(√
2�k(z − t))√2�k
V (t)χk(t) dtdz
∣∣∣∣ ≤1√2‖V ‖1/2L2(0,1).
M(#&pZ+ �J2�(��J2�XA�-0#X2�45+∫ 1
0
[sin(√
2�kz)]2 dz =1
2− sin(2
√2�k)
4√
2�k.
� 'A(U�J+V ≥ 0 7#4H!#+ �'A( +¶73'4XA+J2�@456
1 ≥ A2k(
1
2− 1
4π
)− 2
√2Ak‖V ‖1/2L2(0,1).
`!#'A@�')FTS#X)'A+-@Z45!#2�4�45!#+-"H+ +¶\ 'A@D45@2²(#6,(#(U+J*�2�4H')[�+.�-6,(#@D4�2�(�4C
@H0#��!¢45!32�4
Ak ≤ C(1 + ‖V ‖1/2L2(0,1)), ∀ k ≥ 1.¤=4"H+JFT2�')(#@ 456'A(�%+J�-44H!#'A@XA2�@%4�+-@%4H'AFT2�45+')(6&sM�c ~ +456S#"%6[,+ P9+-FTFT2 M�c c
£6�"H6,X)X2�"DC¬M�cµz¥'A@²2�( +�2�@DC �-6,(#@H+-/10#+J(#�-+�6�:45!#+ �$,XA&U+J"�+-@%4H'AFT2�45+-@ 2�(#&�PK+-FTFT2�M c �')4H!¢4H!#+ "H+JFT2�"%T45!#2�4
�k[V ] − �k[Ṽ ] =∫ 1
0
∂λ�k(λ) dλ =∫ 1
0
〈|χk[W (λ, ·)](z)|2(V − Ṽ )〉 dλ,�!#+J"%+.Z+ +J(U6�45+:L6,"
λ ∈ [0, 1] 7 W (λ, z) = Ṽ + λ(V − Ṽ ) 2�(#& �k(λ) = �k[W (λ, ·)]c
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∀ t ∈ R+, W (t) = W (0) −∫ t
0
D(s) ds ≤ W (0). &sM c >+
M@%@H0#FWSU45')6,( § c .6�(^'A(U')45'A2�XU�4�2�')(#@H0U"H+J@V45!32�4W (0)
'A@VO?6,0U(#+J&f')(#+-S?+-(#+J(�4HX)CW6,(εcv¤a(
:s2��-4J78@H6�X)[E'A(#*T45!#+�E��!#"H$E'A(U*,+J"Q+
M(#& �k[0] = π2k2/2c`!10#@ &sM c >+'AFWS#XA')+J@Z4H!32�4
∑
k
∫
ω
ρk(π2k2
2+ log ρk) dx +
1
2
∫∫
Ω
|∇x,zV |2 dxdz ≤ W (0).&sM c .?+
`v2�E')(#*K =
∑k e
−k2 7#O�Cf45!U+�®�+-(#@H+-(¥')(#+J/1032�X)')4aCW+.!32�[,+
W (0) ≥∑
k
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ω
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∑
k
∫
ω
ρke−k2
(logρk
e−k2)e−k
2
Kdx
≥∫
ω
Ns log(Ns/K) dx.
`!10#@N:L"%6,F &LM�c/.?+¶7
∀ t ∈ R+,∫
ω
Ns log Ns dx ≤ W (0) + NI log K.`!#+J(+-@%4H'AFT2�45+ &YEc §,§ +')@2T')"H+J�¶4�-6,(#@%+J/10#+J(U�J+²6�:45!#+ "H+JFT2�"%^45!32�4 ∀ a > 0 7 a| log a| ≤a log a + 2/e
c¤a(¢:s2��¶4�7
∑
k
∫
ω
ρk| log ρk| dx ≤∑
k
∫
ω
ρk(| log(ρk/e−k2
)| + k2) dx
≤∑
k
∫
ω
ρk(log(ρk/e−k2) + k2) dx +
2
e
∑
k
∫
ω
e−k2
dx
≤∑
k
∫
ω
ρk(log ρk + 2k2) dx +
2
e|ω|K,
�!#+J"%+ |ω| +-(#6�45+-@4H!#+.PK+JOy+J@H*�0#+ FW+�2�@H0#"%+.6�: ω cR`!10#@-7yZ+ +J&U0#�J+ :L"H6,F &sM�c/.?+Z4H!32�4
∀ t ∈ R+,∑
k
∫
ω
ρk| log ρk| dx ≤ W (0) +2
e|ω|K. &LM�cµ?+
�
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∑
k
∫
ω
ρkk2 dx +
1
2
∫∫
Ω
|∇x,zV |2 dxdz ≤ 2W (0) +2
e|ω|K.
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∀ t ∈ R+,∫
ω
|∇x√
Ns|2 dx ≤ C(1 + D(t)),:L6,"92�(#6,(#(#+-*�2�4H')[,+ �-6,(#@%452�(�4v£c�� 'A(#�-++ !32�[,+ :L"H6�F PK+JFWF^2�Ec z 45!#2�4 ∫ t
0D(s) ds ≤ W (0) 7:L6,"2�X)X
t ∈ R+ 7�+�6,O 4�2�')( &Y c § z?+�2�(#&²:L"H6�F 4H!#+ �.2�*,XA'A2�"H�.Uc
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