\documentclass[12pt,reqno]{amsart}
%DIF LATEXDIFF DIFFERENCE FILE
%DIF DEL test2-old.tex   Wed Aug  5 19:23:54 2020
%DIF ADD test2.tex       Wed Aug  5 19:23:10 2020
\usepackage{amssymb,amsfonts,amsmath,amsthm}
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{lem}{Lemma}
\newtheorem{cor}{Corollary}

\newtheorem{prop}{Proposition}

\newtheorem{rem}{Remark}

\newcommand{\setB}{{\mathord{\mathbb B}}}
\newcommand{\setC}{{\mathord{\mathbb C}}}
\newcommand{\setN}{{\mathord{\mathbb N}}}
\newcommand{\setQ}{{\mathord{\mathbb Q}}}
\newcommand{\setR}{{\mathord{\mathbb R}}}
\newcommand{\setT}{{\mathord{\mathbb T}}}
\newcommand{\setZ}{{\mathord{\mathbb Z}}}

\usepackage{paralist}

\def\Id{\mathop{\rm Id}}
% I could add that our page size is 6.38in by 9.82in
%  (with an offset of -1.15in by -0.67in)
  \usepackage{geometry}
  \geometry{totalwidth=5.6in,totalheight=7.48in, layoutvoffset=1.15in%,layouthoffset=0.67in
  }
%% \textheight=9.82in
%% \textwidth=6.38in
%% \voffset=-1.15in
%% \hoffset=-0.67in
\usepackage{setspace}
%\onehalfspacing % or \doublespacing
\setstretch{1.05}
\usepackage{txfonts}
\numberwithin{equation}{section} % get eq numbered within sections
%DIF PREAMBLE EXTENSION ADDED BY LATEXDIFF
%DIF UNDERLINE PREAMBLE %DIF PREAMBLE
\RequirePackage[normalem]{ulem} %DIF PREAMBLE
\RequirePackage{color}\definecolor{RED}{rgb}{1,0,0}\definecolor{BLUE}{rgb}{0,0,1} %DIF PREAMBLE
\providecommand{\DIFadd}[1]{{\protect\color{blue}\uwave{#1}}} %DIF PREAMBLE
\providecommand{\DIFdel}[1]{{\protect\color{red}\sout{#1}}}                      %DIF PREAMBLE
%DIF SAFE PREAMBLE %DIF PREAMBLE
\providecommand{\DIFaddbegin}{} %DIF PREAMBLE
\providecommand{\DIFaddend}{} %DIF PREAMBLE
\providecommand{\DIFdelbegin}{} %DIF PREAMBLE
\providecommand{\DIFdelend}{} %DIF PREAMBLE
\providecommand{\DIFmodbegin}{} %DIF PREAMBLE
\providecommand{\DIFmodend}{} %DIF PREAMBLE
%DIF FLOATSAFE PREAMBLE %DIF PREAMBLE
\providecommand{\DIFaddFL}[1]{\DIFadd{#1}} %DIF PREAMBLE
\providecommand{\DIFdelFL}[1]{\DIFdel{#1}} %DIF PREAMBLE
\providecommand{\DIFaddbeginFL}{} %DIF PREAMBLE
\providecommand{\DIFaddendFL}{} %DIF PREAMBLE
\providecommand{\DIFdelbeginFL}{} %DIF PREAMBLE
\providecommand{\DIFdelendFL}{} %DIF PREAMBLE
%DIF LISTINGS PREAMBLE %DIF PREAMBLE
\RequirePackage{listings} %DIF PREAMBLE
\RequirePackage{color} %DIF PREAMBLE
\lstdefinelanguage{DIFcode}{ %DIF PREAMBLE
%DIF DIFCODE_UNDERLINE %DIF PREAMBLE
  moredelim=[il][\color{red}\sout]{\%DIF\ <\ }, %DIF PREAMBLE
  moredelim=[il][\color{blue}\uwave]{\%DIF\ >\ } %DIF PREAMBLE
} %DIF PREAMBLE
\lstdefinestyle{DIFverbatimstyle}{ %DIF PREAMBLE
	language=DIFcode, %DIF PREAMBLE
	basicstyle=\ttfamily, %DIF PREAMBLE
	columns=fullflexible, %DIF PREAMBLE
	keepspaces=true %DIF PREAMBLE
} %DIF PREAMBLE
\lstnewenvironment{DIFverbatim}{\lstset{style=DIFverbatimstyle}}{} %DIF PREAMBLE
\lstnewenvironment{DIFverbatim*}{\lstset{style=DIFverbatimstyle,showspaces=true}}{} %DIF PREAMBLE
%DIF END PREAMBLE EXTENSION ADDED BY LATEXDIFF

\begin{document}
\begin{align}
  \label{eq:sec3-applications:1}
  A^0(U;t)\partial_t U + \sum_{k=1}^3 A^k(U;t) \partial_k
  U + B(U;t) U&= G(U;t) \\
  \label{eq:sec3-applications:3}
   \Delta\Phi &= 4\pi  R^2 \sigma,
\end{align}


\begin{equation*}
  \label{eq:sec3-applications:15}
  \tag{QSA0}
\begin{cases}
           A^0(U;t)\partial_t U+ \sum_{a=1}^3 A^{a}(U;t)\partial_a U 
           =\widehat G(U;t),\\
           U(0,x)=u_0(x),
         \end{cases}
     \end{equation*}




The main idea of the proof is to show convergence in the
$H^{m-1}(\setT^3)$ norm for the derivatives of the solution. 
So we start by differentiating equation \DIFdelbegin %DIFDELCMD < \eqref{eq:sec3-applications:1} %%%
\DIFdelend \DIFaddbegin \eqref{eq:sec3-applications:15} \DIFaddend with respect to the
spatial variables $D_x$ and obtain

\end{document}
%%% Local Variables:
%%% mode: LaTeX
%%% sentence-end-double-space: nil
%%% TeX-master: t
%%% End: