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We consider the Cauchy problem for the kinetic derivative nonlinearSchr\"odinger equation on the torus: \[ \partial_t u - i \partial_x^2 u =\alpha \partial_x \big( |u|^2 u \big) + \beta \partial_x \big[ H \big( |u|^2\big) u \big] , \quad (t, x) \in [0,T] \times \mathbf{T}, \] where theconstants $\alpha,\beta$ are such that $\alpha \in \mathbf{R}$ 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 Cauchyproblem in Sobolev spaces $H^s$ for $s>3/2$. However, the gauge transformtechnique, which is useful for dealing with the derivative loss in thenonlinearity when $\beta =0$, cannot be directly adapted due to the presence ofthe Hilbert transform. In particular, there has been no result on localwell-posedness in low regularity spaces or global solvability of the Cauchyproblem. In this article, we shall prove local and global well-posedness of theCauchy problem for small initial data in $H^s(\mathbf{T})$, $s>1/2$. To thisend, we make use of the parabolic-type smoothing effect arising from theresonant part of the nonlocal nonlinear term $\beta \partial_x [H(|u|^2)u]$, inaddition to the usual dispersive-type smoothing effect for nonlinearSchr\"odinger equations with cubic nonlinearities. As by-products of the proof,we also obtain smoothing effect and backward-in-time ill-posedness results.

Well-posedness of the Cauchy Problem for the Kinetic DNLS on $\mathbf{T}$