For unclear to me reasons, report
encloses abstract
in a titlepage
environment.
Just patch it to not do it.
\documentclass{report}
\usepackage{etoolbox}
\usepackage{lipsum}
\patchcmd{\abstract}{\titlepage}{\cleardoublepage}{}{}
\patchcmd{\endabstract}{\endtitlepage}{\clearpage}{}{}
\begin{document}
\title{some title}
some introduction
\begin{abstract}
\lipsum[1]
\end{abstract}
\lipsum[2]
\end{document}
The console output shows the page numbering is continuous.
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=pdflatex)
restricted \write18 enabled.
entering extended mode
(./vies.tex
LaTeX2e <2018-12-01>
(/usr/local/texlive/2018/texmf-dist/tex/latex/base/report.cls
Document Class: report 2018/09/03 v1.4i Standard LaTeX document class
(/usr/local/texlive/2018/texmf-dist/tex/latex/base/size10.clo))
(/usr/local/texlive/2018/texmf-dist/tex/latex/etoolbox/etoolbox.sty)
(/usr/local/texlive/2018/texmf-dist/tex/latex/lipsum/lipsum.sty
(/usr/local/texlive/2018/texmf-dist/tex/latex/l3kernel/expl3.sty
(/usr/local/texlive/2018/texmf-dist/tex/latex/l3kernel/expl3-code.tex)
(/usr/local/texlive/2018/texmf-dist/tex/latex/l3kernel/l3pdfmode.def))
(/usr/local/texlive/2018/texmf-dist/tex/latex/l3packages/xparse/xparse.sty)
(/usr/local/texlive/2018/texmf-dist/tex/latex/lipsum/lipsum.ltd.tex))
(./vies.aux) [1{/usr/local/texlive/2018/texmf-var/fonts/map/pdftex/updmap/pdfte
x.map}] [2] [3] (./vies.aux) )</usr/local/texlive/2018/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2018/texmf-dist/fonts/type1/pu
blic/amsfonts/cm/cmr10.pfb>
Output written on vies.pdf (3 pages, 29462 bytes).
Transcript written on vies.log.