# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/NuleO(H);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$Nu\le O(H)$">|; 

$key = q/Hgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$H\gg 1$">|; 

$key = q/E=L^{-d}intfrac{1}{2}|nablau|^2+frac{1}{4}(1-u^2)^2,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$E=L^{-d}\int\frac{1}{2}\vert\nabla u\vert^2+\frac{1}{4}(1-u^2)^2 dx$">|; 

$key = q/uapprox-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$u\approx-1$">|; 

$key = q/partial_xh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\partial_xh$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$u$">|; 

$key = q/EgeO(t^{-1slash3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$E\ge O(t^{-1/3})$">|; 

$key = q/O(t^{1slash3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$O(t^{1/3})$">|; 

$key = q/uapprox1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$u\approx1$">|; 

$key = q/Lgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$L\gg 1$">|; 

$key = q/Nu:=L^{-2}H^{-1}intT,ucdot(1,0,0)dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$Nu:=L^{-2}H^{-1}\int T u\cdot(1,0,0)dx$">|; 

$key = q/lim_{Tuparrowinfty}T^{-1}L^{-1}int_0^Tint_0^L(partial_x^2u)^2dxdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\lim_{T\uparrow\infty}T^{-1}L^{-1}\int_0^T\int_0^L(\partial_x^2u)^2dxdt$">|; 

$key = q/(0,L)^d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$(0,L)^d$">|; 

$key = q/(0,L)^2times(0,H);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$(0,L)^2\times(0,H)$">|; 

$key = q/Tapprox1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$T\approx 1$">|; 

$key = q/Nu=O(H);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$Nu=O(H)$">|; 

$key = q/H,Lgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$H,L\gg 1$">|; 

$key = q/{displaymath}partial_tu+partial_x(frac{1}{2}u^2)+partial_x^2u+partial_x^4u;=;0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="\begin{displaymath}
\partial_tu+\partial_x(\frac{1}{2}u^2)+\partial_x^2u+\partial_x^4u\;=\;0
\end{displaymath}">|; 

$key = q/(0,L);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$(0,L)$">|; 

$key = q/{displaymath}partial_tu+triangle(u(1-u^2)+triangleu);=;0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="\begin{displaymath}
\partial_t u+\triangle(u(1-u^2)+\triangle u)\;=\;0
\end{displaymath}">|; 

$key = q/{displaymath}T;=;1;;mbox{for};z=0,quadT;=;0;;mbox{for};z=H,quadu;=;0;;mbox{for};z=0,H.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="\begin{displaymath}
T\;=\;1\;\;\mbox{for}\;z=0,\quad
T\;=\;0\;\;\mbox{for}\;z=H,\quad
u\;=\;0\;\;\mbox{for}\;z=0,H.
\end{displaymath}">|; 

$key = q/{displaymath}partial_tu+partial_x(frac{1}{2}u^2);=;partial_xf.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="\begin{displaymath}
\partial_tu+\partial_x(\frac{1}{2}u^2)\;=\;\partial_xf.
\end{displaymath}">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$T$">|; 

$key = q/{eqnarraystar}partial_tT+nablacdot(Tu)-triangleT&=&0,-triangleu+nablap&=&T,(0,0,1),nablacdotu&=&0{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="84" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="\begin{eqnarray*}
\partial_tT+\nabla\cdot(Tu)-\triangle T&amp;=&amp;0,\\\\
-\triangle u+\nabla p&amp;=&amp;T (0,0,1),\\\\
\nabla\cdot u&amp;=&amp;0
\end{eqnarray*}">|; 

$key = q/O(t^{-1slash3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$O(t^{-1/3})$">|; 

$key = q/O(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$O(1)$">|; 

1;