Originating with the monograph by Dan Henry (1981), the semigroup approach to evolution problems having a positive sectorial operator in the main part allows us to settle them at various levels of the fractional power scale associated with the main linear operator. This translates into different regularity properties of local solutions to such equations. Specific applications of the abstract results to the {\it 2D surface quasi-geostrophic equation} and the {\it fractional chemotaxis system} will be also presented.
Presentation based on: R. Czaja, T. Dlotko, Sectorial equations on fractional power scales, submitted, January 2023.