In　this　paper　we　study　the　Cauchy　problem　for　one　multidimensional　compressible　nonlocal　model　of　the　dissipative　quasi-geostrophic　equations　and　discuss　the　e　ff　ect　of　the　sign　of　initial　data　on　the　wellposedness　of　this　model.　First,　we　prove　the　existence　and　uniqueness　of　local　smooth　solutions　for　the　Cauchy　problem　for　the　model　with　the　nonnegative　initial　data,　which　seems　to　imply　that　whether　the　well-posedness　of　this　model　holds　or　not　depends　heavily　upon　the　sign　of　the　initial　data　even　for　the　subcritical　case.　Secondly,　for　the　sub-critical　case　1　＜　alpha　＜=　2,　we　obtain　the　global　existence　and　uniqueness　results　of　the　nonnegative　smooth　solution.　Next,　we　prove　the　global　existence　of　the　weak　solution　for　0　＜　alpha　＜=　2　and　upsilon　＞　0.　Finally,　for　the　sub-critical　case　1　＜　alpha　＜=　2,　we　establish　H-beta　(beta　＞=　0)　and　L-p　(p　＞=　2)　decay　rates　of　the　smooth　solution　as　t　-＞　infinity.　A　inequality　for　the　Riesz　transformation　is　also　established.