By cross-correlating the AKARI all-sky survey in six infrared (IR) bands (9, 18, 65, 90, 140 and 160 μm) with the Sloan Digital Sky Survey (SDSS) galaxies, we identified 2357 IR galaxies with a spectroscopic redshift. This is not just one of the largest samples of local IR galaxies, but AKARI provides crucial far-IR (FIR) bands for accurately measuring the galaxy spectral energy distribution (SED) across the peak of the dust emission at > 100 μ m. By fitting modern IR SED models to the AKARI photometry, we measured the total infrared luminosity (L_(IR)) of individual galaxies. Using this L_(IR), we constructed the luminosity functions (LF) of IR galaxies at a median redshift of z= 0.031. The LF agrees well with that at z= 0.0082 (the Revised Bright Galaxy Sample), showing smooth and continuous evolution towards higher redshift LFs measured in the AKARI North Ecliptic Pole (NEP) deep field. By integrating the IR LF weighted by L_(IR), we measured the local cosmic IR luminosity density of Ω_(IR_= (3.8^(+5.8)_(−1.2)) × 10^8 L_⊙ Mpc^(−3). We separate galaxies into active galactic nuclei (AGN), star-forming galaxies (SFG) and composite by using the [N ii]/Hα versus [O iii]/Hβ line ratios. The fraction of AGN shows a continuous increase with increasing L_(IR) from 25 to 90 per cent at 9 10^(11) L_⊙, coinciding with the break of both the SFG and AGN IR LFs. At L_(IR)≤ 10^(11) L_⊙, SFG dominates IR LFs. Only 1.1 ± 0.1 per cent of Ω_(IR) is produced by luminous infrared galaxies (L_(IR) > 10^(11) L_⊙), and only 0.03 ± 0.01 per cent by ultraluminous infrared galaxies (L_(IR) > 10^(12) L_⊙) in the local Universe. Compared with high-redshift results from the AKARI NEP deep survey, we observed a strong evolution of Ω^(SFG)IR^∝ (1 +z)^(4.1±0.4) and Ω^(AGN)IR^∝ (1+z)^(4.1±0.5). Our results show that all of our measured quantities (IR LFs, L^*, Ω^(AGN)IR, Ω^(SFG)IR) show smooth and steady increase from lower redshift (the Revised Bright Galaxy Sample) to higher redshift (the AKARI NEP deep survey)
