\newcommand{\dt}{\Delta t}
\newcommand{\uint}{\widetilde{\mathbf{u}}}
\newcommand{\uu}{\mathbf{u}}
\newcommand{\mO}{\mathcal{O}}
\newcommand{\TT}{\mbox{\scriptsize T}}
\def\Div{{\rm div}\,}
\newcommand{\bu} {\mathbf{u}}
\newcommand{\bv} {\mathbf{v}}
\newcommand{\bw} {\mathbf{w}}
\newcommand{\bx} {\mathbf{x}}
\newcommand{\blf} {\mathbf{f}}
\newcommand{\bg} {\mathbf{g}}
\newcommand{\QQ} {\mathbb{Q}}
\newcommand{\VV} {\mathbb{V}}
\newcommand{\bH} {\mathbf{H}}
\newcommand{\dO} {{\partial\Omega}}
\newcommand{\cA} {{\cal A}}
\newcommand{\cB} {{\cal B}}
\newcommand{\Lc} {{\cal L}}
\newcommand{\Frac}[2]{\displaystyle{\frac{#1}{#2}}}
\newcommand{\pder}[2]{\Frac{\partial #1}{\partial #2}}
\def\epsilon{{\varepsilon}}
\def\tq{ {\tilde q}}
\def\tp{ {\tilde p}}
\def\tbv{ {\tilde \bv}}
\def\tbu{ {\tilde \bu}}
\def\cdt{{\hskip-0.08ex\cdot\hskip-0.08ex}}
\newcommand{\vtau}{\mbox{\boldmath$\tau$\unboldmath}}
\newcommand{\vel}{\mathbf{u}}
\newcommand{\pre}{P}
%\newcommand{\Div}[1]{\nabla \cdot {#1}}
%\newcommand{\Frac}[2]{\displaystyle{\frac{#1}{#2}}}
\newcommand{\Deriv}[3]{\Frac{\partial^{#1} #2}{(\partial #3)^{#1}}}
\newcommand{\der}[2]{\frac{d #1}{d #2}}
\newcommand{\Innersp}[2]{\left( #1, #2 \right)_{{\cal H},\Omega}}
%\newcommand{\matA}{{\rm A}}
%\newcommand{\matB}{{\rm B}}
%\newcommand{\matC}{{\rm C}}
%\newcommand{\matD}{{\rm D}}
%\newcommand{\matE}{{\rm E}}
%\newcommand{\matF}{{\rm F}}
%\newcommand{\matG}{{\rm G}}
%\newcommand{\matH}{{\rm H}}
%\newcommand{\matL}{{\rm L}}
%\newcommand{\matR}{{\rm R}}
%\newcommand{\matZ}{{\rm Z}}
%\newcommand{\matM}{{\rm M}}
%\newcommand{\matN}{{\rm N}}
%\newcommand{\matK}{{\rm K}}
%\newcommand{\matS}{{\rm S}}
%\newcommand{\matI}{{\rm I}}
%\newcommand{\matQ}{{\rm Q}}
%\newcommand{\matT}{{\rm T}}
%\newcommand{\matV}{{\rm V}}
%\newcommand{\matU}{{\rm U}}
\newcommand{\matA}{{ A}}
\newcommand{\matB}{{ B}}
\newcommand{\matC}{{ C}}
\newcommand{\matD}{{ D}}
\newcommand{\matE}{{ E}}
\newcommand{\matF}{{ F}}
\newcommand{\matG}{{ G}}
\newcommand{\matH}{{ H}}
\newcommand{\matL}{{ L}}
\newcommand{\matR}{{ R}}
\newcommand{\matZ}{{ Z}}
\newcommand{\matM}{{ M}}
\newcommand{\matN}{{ N}}
\newcommand{\matK}{{ K}}
\newcommand{\matS}{{ S}}
\newcommand{\matI}{{ I}}
\newcommand{\matQ}{{ Q}}
\newcommand{\matT}{{ T}}
\newcommand{\matV}{{ V}}
\newcommand{\matU}{{ U}}

\newcommand{\normld}[2]{||#1||_{{\it l}^2(#2)}}
\newcommand{\normli}[2]{||#1||_{{\it l}^\infty(#2)}}


\newcommand{\ds}{\displaystyle}

% some definitions of bold math italics to make typing easier.
% They are used in the corollary.

\def\bfE{\mbox{\boldmath$E$}}
\def\bfG{\mbox{\boldmath$G$}}

\newcommand{\EE}{\mathbf{E}}
\newcommand{\ee}{\mathrm{e}}

%\newtheorem{thm}{Theorem}
%\newtheorem{lem}{Lemma}
%\newtheorem{prop}{Proposition}

