Large data global well-posedness for the modified Novikov-Veselov system

TL;DR

This work proves global well-posedness and scattering for the two-dimensional, $L^2$-critical modified Novikov-Veselov (mNV) equation at large data using inverse scattering and a nonlinear Gagliardo–Nirenberg inequality for the scattering transform. The analysis hinges on a delicate combination of the DS II scattering framework, time-divisible Strichartz-type norms, and a refined GN bound that controls the nonlinear evolution in $L^2$-based spaces; these ideas yield global existence, dispersive bounds, and asymptotic completeness with wave operators that are smooth diffeomorphisms of $L^2$. The paper also derives a global NV theory for data in the Miura-range, fully characterized by a sharp AAP-type criterion: $H_q=-\Delta+q\ge 0$ if and only if $q$ lies in the Miura image of $L^2$, with the Miura inverse continuous on this range. Collectively, the results connect integrable transform techniques with nonlinear PDE methods to address large-data dynamics in 2D dispersive flows, and extend the scope of global well-posedness and scattering in this critical regime.

Abstract

The modified Novikov-Veselov system (mNV) is a cubic third order dispersive evolution in two space dimensions. It is also completely integrable, belonging to the same hierarchy as the defocusing Davey-Stewartson II (DS II) system. The mNV system is $L^2$ critical. Some time ago, Schottdorf proved that for small $L^2$ initial data, the mNV equation is globally well-posed. In this article, we consider instead the large data problem, using inverse scattering methods. Our main result asserts that the mNV system is globally well-posed for large $L^2$ data, with the solutions scattering as time goes to $\pm \infty$. One key ingredient in the proof, which is of independent interest, is a new nonlinear Gagliardo-Nirenberg inequality for the associated scattering transform. As a byproduct of our main result, we are also able to prove a global well-posedness result for the closely related Novikov-Veselov problem at the critical $\dot H^{-1} + L^1$ level, for a range of data which can heuristically be described as soliton-free. Here we use the associated Miura map to connect the mNV and the NV flows. In order to characterize the range of the Miura map, we prove another result of independent interest, namely a sharp, scale invariant form of the Agmon-Allegretto-Piepenbrink principle in the critical case of two space dimensions.

Theorems & Definitions (43)

• Theorem 1.1: Nachman-Regev-Tataru NRT2020
• Theorem 1.2: Nachman-Regev-Tataru NRT2020
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Proposition 2.2
• Proposition 2.3
• Remark 3.1
• Theorem 3.2