In　this　paper　we　consider　some　equations　similar　to　Navier-Stokes　equations,　the　three-dimensional　Leray-alpha　equations　with　space　periodic　boundary　conditions.　We　establish　the　regularity　of　the　equations　by　using　the　classical　Faedo-Galerkin　method.　Our　argument　shows　that　there　exist　an　unique　weak　solution　and　an　unique　strong　solution　for　all　the　time　for　the　Leray-alpha　equations,　furthermore,　the　strong　solutions　are　analytic　in　time　with　values　in　the　Gevrey　class　of　functions　(for　the　space　variable).　The　relations　between　the　Leray-alpha　equations　and　the　Navier-Stokes　equations　are　also　considered.　(c)　2004　Elsevier　Inc.　All　rights　reserved.