Global well-posedness of the periodic Majda-Biello systems in the resonant case
Abstract
We study the Cauchy problem for the following Majda-Biello system in the resonant case Ξ±β (0,4]β{1} on the periodic domain π. { rcl ut + uxxx & = & β v vx, vt + Ξ±vxxx & = & β (uv)x, (u,v)|t=0 & = & (u0,v0), . x β π, t β β. Here, the initial data u0 and v0 live in the Sobolev space Hs(π) and u0 has zero mean over π. We establish the optimal global well-posedness for the above system when sβ₯ min{1, sΞ±+}, where sΞ± is a constant depending only on Ξ±, and sΞ±= 1/2 for almost every Ξ±β (0,4]β{1}. These indices match the sharp ones in the local theory obtained by Oh (Int. Math. Res. Not., 18:3516-3556, 2009). The main novelty of our proof is to introduce a third modified energy in the I-method by performing a partial correction to the second modified energy around the resonant region. This partial correction avoids the singularity in the full correction, and therefore makes it possible to pointwise bound the quintilinear multipliers arising in the time derivative of the third modified energy. Another feature in our proof is the combination of the Diophantine approximation theory with the sparsity of nearly resonant frequency pairs on the periodic lattice to obtain the key quadrilinear and quintilinear estimates, leading to an almost conservation law sufficient for the global iteration.
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BibTeX
@misc{wang2026global,
title = {{Global well-posedness of the periodic Majda-Biello systems in the resonant case}},
author = {Ke Wang and Xin Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22260},
eprint = {2610.22260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22260},
}
RIS
TY - UNPB
TI - Global well-posedness of the periodic Majda-Biello systems in the resonant case
AU - Ke Wang
AU - Xin Yang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.22260
UR - https://arxiv.org/abs/2610.22260
AB - We study the Cauchy problem for the following Majda-Biello system in the resonant case Ξ±β (0,4]β{1} on the periodic domain π. { rcl u_t + u_xxx & = & β v v_x, v_t + Ξ±v_xxx & = & β (uv)_x, (u,v)|_t=0 & = & (u_0,v_0), . x β π, t β β. Here, the initial data u_0 and v_0 live in the Sobolev space H^s(π) and u_0 has zero mean over π. We establish the optimal global well-posedness for the above system when sβ₯ min{1, s_Ξ±+}, where s_Ξ± is a constant depending only on Ξ±, and s_Ξ±= 1/2 for almost every Ξ±β (0,4]β{1}. These indices match the sharp ones in the local theory obtained by Oh (Int. Math. Res. Not., 18:3516-3556, 2009). The main novelty of our proof is to introduce a third modified energy in the I-method by performing a partial correction to the second modified energy around the resonant region. This partial correction avoids the singularity in the full correction, and therefore makes it possible to pointwise bound the quintilinear multipliers arising in the time derivative of the third modified energy. Another feature in our proof is the combination of the Diophantine approximation theory with the sparsity of nearly resonant frequency pairs on the periodic lattice to obtain the key quadrilinear and quintilinear estimates, leading to an almost conservation law sufficient for the global iteration.
ER -