Below there's one possible solution; the key ideas were:
I used the solution given in How to keep a constant baselineskip when using minipages (or \parboxes)? to guarantee spacing after the minipages.
Before the minipages I used \smallskip\nointerlinespacing.
I defined a newenvironment sminipage (to apply \small inside minipage); the optional argument (set by default to t) controls the alignment of the minipage and the mandatory argument sets the width of the minipage.
I also made some other modifications to improve your code:
I removed the spurious blank space after the first \end{minipage}.
I defined a \Pder command to facilitate the writing of the partial derivatives.
The idea was to obtain (approximately) the same spacing around the equations inside the minipage as the one used for regular equations not inside a minipage (I added a regular equation environment at the end just for comparison purposes):
\documentclass{article}
\usepackage{amsmath}
\usepackage[nopar]{lipsum}% just to generate text for the example
\newcommand\Pder[2]{%
\frac{\mathrm{d}#1}{\mathrm{d}#2}}
\newenvironment{sminipage}[2][t]
{\minipage[t]{#2}\small}
{\endminipage}
\begin{document}
\lipsum*[4]\par\smallskip\nointerlineskip
\noindent\begin{sminipage}[t]{0.5\textwidth}
\begin{equation}\label{e3}
\begin{split}
\Pder{x'}{t} &= \Pder{x}{t} - \Pder{\phantom{x}}{t} (ut) \\
\Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{\phantom{x}}{t} (t) \\
\Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{t}{t} \\
v' &= v - u
\end{split}
\end{equation}
\end{sminipage}%
\begin{sminipage}{0.5\textwidth}
\begin{equation}\label{e4}
\begin{split}
\Pder{x}{t} &= \Pder{x'}{t} - \Pder{\phantom{x}}{t} (ut) \\
\Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{\phantom{x}}{t} (t) \\
\Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{t}{t} \\
v &= v' + u
\end{split}
\end{equation}\null
\par\xdef\tpd{\the\prevdepth}
\end{sminipage}
\prevdepth\tpd
\noindent\lipsum[2]
\begin{equation}
a=b
\end{equation}
\lipsum[4]
\end{document}

I defined now (as was requested in a comment) a new environment eqmpage which basically is a top aligned minipage with constant width of \linewidth which automates all the preparations mentioned above:
\documentclass{article}
\usepackage{amsmath}
\usepackage[nopar]{lipsum}% just to generate text for the example
\newcommand\Pder[2]{%
\frac{\mathrm{d}#1}{\mathrm{d}#2}}
\newenvironment{sminipage}[2][t]
{\minipage[t]{#2}\small}
{\endminipage}
\newenvironment{eqmpage}
{\par\smallskip\nointerlineskip%
\noindent\minipage[t]{\textwidth}}
{\par\xdef\tpd{\the\prevdepth}\endminipage\par\prevdepth\tpd}
\begin{document}
\lipsum*[4]
\begin{eqmpage}
\begin{sminipage}[t]{0.5\textwidth}
\begin{equation}\label{e3}
\begin{split}
\Pder{x'}{t} &= \Pder{x}{t} - \Pder{\phantom{x}}{t} (ut) \
\Pder{x'}{t} &= \Pder{x}{t} - u,\Pder{\phantom{x}}{t} (t) \
\Pder{x'}{t} &= \Pder{x}{t} - u,\Pder{t}{t} \
v' &= v - u
\end{split}
\end{equation}
\end{sminipage}%
\begin{sminipage}{0.5\textwidth}
\begin{equation}\label{e4}
\begin{split}
\Pder{x}{t} &= \Pder{x'}{t} - \Pder{\phantom{x}}{t} (ut) \
\Pder{x}{t} &= \Pder{x'}{t} - u,\Pder{\phantom{x}}{t} (t) \
\Pder{x}{t} &= \Pder{x'}{t} - u,\Pder{t}{t} \
v &= v' + u
\end{split}
\end{equation}\null
\end{sminipage}
\end{eqmpage}
\noindent\lipsum[2]
\begin{equation}
a=b
\end{equation}
\lipsum[4]
\end{document}