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We update our recently published model for GAlaxy Evolution and Assembly(GAEA), to include a self-consistent treatment of the partition of cold gas inatomic and molecular hydrogen. Our model provides significant improvements withrespect to previous ones used for similar studies. In particular, GAEA (i)includes a sophisticated chemical enrichment scheme accounting fornon-instantaneous recycling of gas, metals, and energy; (ii) reproduces themeasured evolution of the galaxy stellar mass function; (iii) reasonablyreproduces the observed correlation between galaxy stellar mass and gasmetallicity at different redshifts. These are important prerequisites formodels considering a metallicity dependent efficiency of molecular gasformation. We also update our model for disk sizes and show that modelpredictions are in nice agreement with observational estimates for the gas,stellar and star forming disks at different cosmic epochs. We analyse theinfluence of different star formation laws including empirical relations basedon the hydro-static pressure of the disk, analytic models, and prescriptionsderived from detailed hydro-dynamical simulations. We find that modifying thestar formation law does not affect significantly the global properties of modelgalaxies, neither their distributions. The only quantity showing significantdeviations in different models is the cosmic molecular-to-atomic hydrogenratio, particularly at high redshift. Unfortunately, however, this quantityalso depends strongly on the modelling adopted for additional physicalprocesses. Useful constraints on the physical processes regulating starformation can be obtained focusing on low mass galaxies and/or at higherredshift. In this case, self-regulation has not yet washed out differencesimprinted at early time.

H$_2$-based star formation laws in hierarchical models of galaxy formation