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

doi:10.1007/s10473-023-0517-5

Abstract

Abstract: We consider the fourth-order nonlinear Schrödinger equation (4NLS) \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 $H^s(\mathbb{R}^d)$ for $s\in (s_c-1/2, \ s_c]$, when $d\geq3$ and $m\geq5$, where $s_c:=d/2-2/(m-1)$ is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new $M$-norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for $d=2, \ m\geq4$ and $ 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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References

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

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