Finite-time Langmuir collapse for the three-dimensional Zakharov system
Consider the Zakharov system on $\mathbb{R}^3$, with unit ion sound speed, $$i \partial_t u - \Delta u = n u, \qquad \partial_{tt} n - \Delta n = -\Delta |u|^2, \qquad (u, n, \partial_t n)|_{t=0} = (u_0, n_0, n_1),$$ for $u$ complex and $n$ real, introduced by Zakharov as a model of Langmuir turbulence in an unmagnetized plasma. It conserves the mass $\|u\|_{L^2}^2$ and an energy, and Bourgain and Colliander proved local well-posedness in the energy space $H^1 \times L^2 \times \dot H^{-1}$. Merle proved that radial solutions of negative energy blow up in finite or infinite time, meaning that the energy-space norm is unbounded as $t$ approaches the end of the maximal interval. Guo, Nakanishi and Wang showed that radial solutions whose energy times mass is below that of the ground-state standing wave either scatter or blow up in finite or infinite time, and Guo and Nakanishi classified the radial solutions with energy slightly above the ground-state energy into nine regions, according to whether they scatter, grow up (in finite or infinite time) or stay near the ground states, forward and backward in time. In two dimensions Glangetas and Merle constructed self-similar solutions blowing up in finite time, and for the three-dimensional Klein–Gordon–Zakharov system with ion sound speed $\ne 1$, Guo, Nakanishi and Wang proved that radial data below the ground-state energy either scatter or blow up in finite time.
Problem. Do there exist smooth rapidly decaying initial data $u_0 : \mathbb{R}^3 \to \mathbb{C}$ and $n_0, n_1 : \mathbb{R}^3 \to \mathbb{R}$ for which the solution has a finite maximal energy-space existence time $T_+ < \infty$ and $$\limsup_{t \uparrow T_+} \Bigl( \|u(t)\|_{H^1} + \|n(t)\|_{L^2} + \|\partial_t n(t)\|_{\dot H^{-1}} \Bigr) = \infty?$$ Here $\|g\|_{\dot H^{-1}} = \||D|^{-1} g\|_{L^2}$ and all norms are on $\mathbb{R}^3$. Unbounded growth only as $t \to \infty$ does not answer the question.
Source: Zakharov (1972), who predicted finite-time collapse of Langmuir waves in three dimensions; recorded as an open mathematical problem by Guo, Nakanishi and Wang (2014), introduction, and Guo and Nakanishi (2026), Remark 1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, blow-up, Langmuir waves, Zakharov system
References
@article{guo2014kgz,
author = {Guo, Zihua and Nakanishi, Kenji and Wang, Shuxia},
title = {Global dynamics below the ground state energy for the {K}lein--{G}ordon--{Z}akharov system in the {3D} radial case},
journal = {Comm. Partial Differential Equations},
volume = {39},
number = {6},
year = {2014},
pages = {1158--1184},
doi = {10.1080/03605302.2013.836715},
eprint = {1304.3570},
archivePrefix = {arXiv},
}
@misc{guo2026,
author = {Guo, Zihua and Nakanishi, Kenji},
title = {Global dynamics above the ground state energy for the {3D} {Z}akharov system},
year = {2026},
eprint = {2608.14993},
archivePrefix = {arXiv},
}
@article{zakharov1972,
author = {Zakharov, V. E.},
title = {Collapse of {L}angmuir waves},
journal = {Sov. Phys. JETP},
volume = {35},
year = {1972},
pages = {908--914},
}
@article{bourgain1996,
author = {Bourgain, J. and Colliander, J.},
title = {On wellposedness of the {Z}akharov system},
journal = {Int. Math. Res. Not. IMRN},
volume = {1996},
number = {11},
year = {1996},
pages = {515--546},
doi = {10.1155/S1073792896000359},
}
@article{merle1996,
author = {Merle, Frank},
title = {Blow-up results of virial type for {Z}akharov equations},
journal = {Comm. Math. Phys.},
volume = {175},
number = {2},
year = {1996},
pages = {433--455},
doi = {10.1007/BF02102415},
}
@article{guo2013zakharov,
author = {Guo, Zihua and Nakanishi, Kenji and Wang, Shuxia},
title = {Global dynamics below the ground state energy for the {Z}akharov system in the {3D} radial case},
journal = {Adv. Math.},
volume = {238},
year = {2013},
pages = {412--441},
doi = {10.1016/j.aim.2013.02.008},
}
@article{glangetas1994,
author = {Glangetas, L. and Merle, F.},
title = {Existence of self-similar blow-up solutions for {Z}akharov equation in dimension two. {I}},
journal = {Comm. Math. Phys.},
volume = {160},
number = {1},
year = {1994},
pages = {173--215},
doi = {10.1007/BF02099792},
}
@misc{aim-zakharov-three-dimensional-collapse,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite-time Langmuir collapse for the three-dimensional Zakharov system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/356-zakharov-three-dimensional-collapse.md},
note = {Entry 356 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00657, "Finite-time Langmuir collapse for the three-dimensional Zakharov system", clopen.solutions.