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By cross‐correlating the  AKARI  all‐sky survey in six infrared (IR) bands (9, 18, 65, 90, 140 and 160  μ m) 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  &gt; 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 ]/H α  versus [O  iii ]/H β  line ratios. The fraction of AGN shows a continuous increase with increasing   L  IR   from 25 to 90 per cent at  9 &lt; log  L  IR  &lt; 12.5 . The SFR  Hα   and  show good correlations with   L  IR   for SFG and AGN, respectively. The self‐absorption‐corrected H α /H β  ratio shows a weak increase with   L  IR   with a substantial scatter. When we separate IR LFs into contributions from AGN and SFG, the AGN contribution becomes dominant at   L  IR  &gt; 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  &gt; 10 11  L ⊙  ), and only 0.03  ±  0.01 per cent by ultraluminous infrared galaxies (  L  IR  &gt; 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).

Infrared luminosity functions of AKARI Sloan Digital Sky Survey galaxies