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

Abstract

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus: \[ \partial_t u - i \partial_x^2 u = α\partial_x \big( |u|^2 u \big) + β\partial_x \big[ H \big( |u|^2 \big) u \big] , \quad (t, x) \in [0,T] \times \mathbf{T}, \] where the constants $\alpha ,\beta$ 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 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 article, we shall prove local and global well-posedness of the Cauchy problem for small initial data in $H^s(\mathbf{T})$, $s>1/2$. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term $\beta {\partial }_x[H(|u{|}^2)u]$, 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 smoothing effect and backward-in-time ill-posedness results.

version 2: the introduction slightly expanded and related references added, some modification in the appendix. 49 pages

References in corpus (2)

• $H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation
• Global well-posedness for the derivative nonlinear Schrödinger equation