We consider a certain nonlinear Schrodinger equation (NLS) with nonlocal derivative nonlinearity, which is called the kinetic derivative NLS (KDNLS). The Cauchy problem for the standard derivative NLS has been proved to be locally well-posed in low regularity by the gauge transform or by its complete integrability, but these techniques do not work well for KDNLS. On the other hand, KDNLS has dissipative nature, and especially in the periodic setting, a first-order parabolic term arises from the resonant nonlinear interactions. Taking advantage of this parabolic structure, we prove small-data local and global well-posedness results for periodic KDNLS in low-regularity Sobolev spaces. This is a joint work with Yoshio Tsutsumi (Kyoto University).