Local well-posedness for the dispersion generalized periodic KdV equation

In this paper, we set up the local well-posedness of the initial value problem for the dispersion generalized periodic KdV equation: ∂ t u + ∂ x | D x | α u = ∂ x u 2 , u ( 0 ) = φ for α > 2 , s ⩾ − α 4 and φ ∈ H s ( T ) . And we show that the − α 4 is a lower endpoint to obtain the bilinear estimates (1.2) and (1.3) which are the crucial steps to obtain the local well-posedness by Picard iteration. The case α = 2 was studied in Kenig et al. (1996) [10].