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The variation of nutrient supply not only leads to the differences in the phytoplankton biomass and primary productivity but also induces the long-term phenotypic evolution of phytoplankton. It is widely accepted that marine phytoplankton follows Bergmann's Rule and becomes smaller with climate warming. Compared with the direct effect of increasing temperature, the indirect effect via nutrient supply is considered to be an important and dominant factor in the reduction of phytoplankton cell size. In this paper, a size-dependent nutrient-phytoplankton model is developed to explore the effects of nutrient supply on the evolutionary dynamics of functional traits associated with phytoplankton size. The ecological reproductive index is introduced to investigate the impacts of input nitrogen concentration and vertical mixing rate on the persistence of phytoplankton and the distribution of cell size. In addition, by applying the adaptive dynamics theory, we study the relationship between nutrient input and the evolutionary dynamics of phytoplankton. The results show that input nitrogen concentration and vertical mixing rate have significant effects on the cell size evolution of phytoplankton. Specifically, cell size tends to increase with the input nutrient concentration, as does the diversity of cell sizes. In addition, a single-peaked relationship between vertical mixing rate and cell size is observed. When the vertical mixing rate is too low or too high, only small individuals are dominant in the water column. When the vertical mixing rate is moderate, large individuals can coexist with small individuals, so the diversity of phytoplankton is elevated. We predict that reduced intensity of nutrient input due to climate warming will lead to a trend towards smaller cell size and will reduce the diversity of phytoplankton.