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The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities

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Shaopeng Xu is supported by Hainan Provincial Natural Science Foundation of China (No.2019RC168)

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  • Chen and Zhang [7] consider the probabilistic Cauchy problem of the fourth order Schrödinger equation

    $ \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u = P_m((\partial_x^\alpha u)_{|\alpha|\leq2},(\partial_x^\alpha \overline{u})_{|\alpha|\leq2}),\ m\geq3, \end{align*} $

    where $ P_m $ is a homogeneous polynomial of degree $ m $. The almost sure local well-posedness and small data global existence were obtained in $ H^s(\mathbb{R}^d) $ with the regularity threshold $ s_c-1/2 $ when $ d\geq3 $, where $ s_c: = d/2-2/(m-1) $ is the scaling critical regularity. For the lower regularity threshold $ (d-1)s_c/d $ with $ m = 2 $ and $ s_c-\min\{1,d/4\} $ with $ m\geq3 $, we get the corresponding well-posedness of the following fourth order nonlinear Schrödinger equation

    $ \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u = P_m((\partial_x^\alpha \overline{u})_{|\alpha|\leq2}),\ m\geq2 \end{align*} $

    on $ {\mathbb{R}}^d $ ($ d\geq2 $) with random initial data.

    Mathematics Subject Classification: Primary: 35Q41, 35Q55; Secondary: 35R60, 60H15.


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