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Almost Sure Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Large Initial Data

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Abstract

We consider the fourth-order nonlinear Schrödinger equation (4NLS)

$$({\rm{i}}{\partial _t} + \varepsilon \Delta + {\Delta ^2})u = {c_1}{u^m} + {c_2}(\partial u){u^{m - 1}} + {c_3}{(\partial u)^2}{u^{m - 2}},$$

and establish the conditional almost sure global well-posedness for random initial data in \({H^s}({\mathbb{R}^d})\) for s ∈ (sc − 1/2, sc], when d ≥ 3 and m ≥ 5, where sc:= d/2 − 2/(m − 1) is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new M-norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for d = 2, m ≥ 4 and d ≥ 3, 3 ≤ m < 5.

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Authors and Affiliations

  1. Department of Mathematics, **an University, Guangzhou, 510632, China

    Mingjuan Chen

  2. School of Mathematical Sciences, Peking University, Bei**g, 100871, China

    Shuai Zhang

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  1. Mingjuan Chen
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  2. Shuai Zhang
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Correspondence to Shuai Zhang.

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Conflict of Interest

The authors declare no conflict of interest.

Chen’s research was supported by the National Natural Science Foundation of China (12001236), the Natural Science Foundation of Guangdong Province (2020A1515110494).

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Chen, M., Zhang, S. Almost Sure Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Large Initial Data. Acta Math Sci 43, 2215–2233 (2023). https://doi.org/10.1007/s10473-023-0517-5

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  • DOI: https://doi.org/10.1007/s10473-023-0517-5

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