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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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    2

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    ~

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    ')@+ 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

    dt

    k

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    2

    d

    dt

    ∫∫

    |∇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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    H101O�C

    J(V ) =1

    2

    ∫∫

    |∇V |2 dxdz +∫

    ω

    Ns log∑

    k

    e−�k[Rε[V ]] dx = J0(V ) + J1(V, Ns).

    &szUc § ?+

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    |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�(#&

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    'AFWS#X)C

    ω

    Ns‖Rε[V − Ṽ ]‖L2z(0,1) dx ≤ CT‖Rε[V − Ṽ ]‖H1(Ω) ≤ CT‖V − Ṽ ‖H1(Ω),&szUc § .?+

  • 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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    "9�O'Ñs

    "94K�L9�.89��� �.�

    L∞(0, T ; L1(ω))"9�O') U"9���!

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    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#��!

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