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• W3197804530 abstract "We consider the Cauchy problem for the kinetic derivative nonlinear Schrodinger 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 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 Schrodinger equations with cubic nonlinearities. As by-products of the proof, we also obtain smoothing effect and backward-in-time ill-posedness results." @default.
• W3197804530 created "2021-09-13" @default.
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• W3197804530 date "2021-08-30" @default.
• W3197804530 modified "2023-09-26" @default.
• W3197804530 title "Well-posedness of the Cauchy Problem for the Kinetic DNLS on $mathbf{T}$" @default.
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• W3197804530 doi "https://doi.org/10.48550/arxiv.2108.13001" @default.
• W3197804530 hasPublicationYear "2021" @default.
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