ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION WITH LARGE INITIAL DATA*

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doi:10.1007/s10473-023-0517-5

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

\begin{aligned} (i \partial_{t} + ε Δ + Δ^{2}) u = c_{1} u^{m} + c_{2} (\partial u) u^{m - 1} + c_{3} (\partial u)^{2} u^{m - 2}, \end{aligned}

$\begin{align*} ({\rm i}\partial_t+\varepsilon\Delta+\Delta^2)u=c_1u^m+c_2(\partial u)u^{m-1}+c_3(\partial u)^2u^{m-2}, \end{align*}$ and establish the conditional almost sure global well-posedness for random initial data in

H^{s} (R^{d})

$H^s(\mathbb{R}^d)$ for

s \in (s_{c} - 1 / 2, s_{c}]

$s\in (s_c-1/2, \ s_c]$ , when

d \geq 3

$d\geq3$ and

m \geq 5

$m\geq5$ , where

s_{c} := d / 2 - 2 / (m - 1)

$s_c:=d/2-2/(m-1)$ is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new

M

$M$ -norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for

d = 2, m \geq 4

$d=2, \ m\geq4$ and

d \geq 3, 3 \leq m < 5

$d\geq3, \ 3\leq m<5$ .

Key words: fourth-order Schrödinger equation, random initial data, almost sure global well-posedness, $M$ -norm, stability theory

CLC Number:

35Q55

Mingjuan Chen, Shuai Zhang. ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION WITH LARGE INITIAL DATA^*[J].Acta mathematica scientia,Series B, 2023, 43(5): 2215-2233.

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