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

We consider the Cauchy problem for the kinetic derivative nonlinear Schrodinger equation on the torus T = R/2 pi Z : partial derivative(t)u - i partial derivative(2)(x)u = alpha partial derivative(x)(|u|(2)u) + beta partial derivative(x)[H(|u|(2))u] for (t, x) is an element of [0, T] x T, where the constants alpha, beta are such that alpha is an element of R and beta < 0, and H 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 H-s for s > 3/2. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when beta = 0, 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 H-s(T), s > 1/2. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term beta partial derivative(x)[H(|u|(2))u], in addition to the usual dispersive-type smoothing effect for nonlinear Schrodinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.