Well-posedness of the Cauchy problem for the kinetic DNLS on T

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus [Formula: see text] for [Formula: see text], where the constants [Formula: see text] are such that [Formula: see text] and [Formula: see text], and [Formula: see text] denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces [Formula: see text] for [Formula: see text]. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when [Formula: see text], cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in [Formula: see text], [Formula: see text]. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term [Formula: see text], in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.