This　paper　considers　the　-stability　results　for　the　first　order　fully　discrete　schemes　based　on　the　mixed　finite　element　method　for　the　time-dependent　Navier-Stokes　equations　with　the　initial　data　with　and　2.　A　mixed　finite　element　method　is　used　to　the　spatial　discretization　of　the　Navier-Stokes　equations,　and　the　temporal　treatments　of　the　spatial　discrete　Navier-Stokes　equations　are　the　first　order　implicit,　semi-implicit,　implicit/explicit(the　semi-implicit/explicit　in　the　case　of　)　and　explicit　schemes.　The　-stability　results　of　the　schemes　are　provided,　where　the　first　order　implicit　and　semi-implicit　schemes　are　the　-unconditional　stable,　the　first　order　explicit　scheme　is　the　-conditional　stable,　and　the　implicit/explicit　scheme　(the　semi-implicit/explicit　scheme　in　the　case　of　)　is　the　-almost　unconditional　stable.　Moreover,　this　paper　makes　some　numerical　investigations　of　the　-stability　results　for　the　first　order　fully　discrete　schemes　for　the　time-dependent　Navier-Stokes　equations.　Through　a　series　of　numerical　experiments,　it　is　verified　that　the　numerical　results　are　shown　to　support　the　developed　-stability　theory.