Sharp polynomial growth of higher Sobolev norms for NLS on irrational 3-tori: the irrationality gain, its Diophantine hypotheses, and the energy-critical endpoint

For the defocusing nonlinear Schrödinger equation $(i\partial_t+\Delta)u=|u|^{p-1}u$ on a three-dimensional rectangular torus $\mathbb{T}^3_\ell$ with super-cubic sub-quintic nonlinearity ($3<p<5$), Deng and Germain proved that, for generic (full-measure) side lengths $\ell$, the $H^2$ norm grows at most like $(1+|t|)^{2/(5-p+\theta(p))}$ with $\theta(p)=\min(p-3,5-p)/182$, improving on the rational-torus-type exponent $2/(5-p)$; they stated that their choice of $\theta(p)$ is far from optimal and that only sufficient Diophantine conditions on $\ell$ are known. The open problem is to sharpen and complete this picture: quantify the optimal irrationality gain in the polynomial growth exponent for higher Sobolev norms, characterize the sharp arithmetic (Diophantine) conditions on the torus under which the improvement holds, extend the improved bounds to Sobolev indices $s>2$ for the unregularized nonlinearity, and treat the energy-critical endpoint $p=5$ beyond the small-energy regime.

Background

The nonlinear Schrödinger equation (NLS) is the dispersive PDE $(i\partial_t+\Delta)u=\pm|u|^{p-1}u$; the sign $+$ (as here) is called defocusing and makes the conserved energy positive definite. On the rectangular torus $\mathbb{T}^3_\ell=[0,\ell_1]\times[0,\ell_2]\times[0,\ell_3]$ with energy $E_\ell[u]=\int_{\mathbb{T}^3_\ell}\tfrac{1}{2}|\nabla u|^2+\tfrac{1}{p+1}|u|^{p+1}\,dx$, solutions propagate regularity: data in a Sobolev space $H^s$ (the space of functions with $s$ square-integrable derivatives) remain in $H^s$ for all time when $3\le p<5$. A torus is called rational when the ratios of the $\ell_i$ are rational and irrational otherwise; irrational tori are generic (a full-measure set among side lengths). A recurring question for Hamiltonian PDE is how fast $\|u(t)\|_{H^s}$ may grow while the $H^1$-level energy stays bounded — the mathematical signature of weak turbulence, i.e. the transfer of energy toward higher and higher frequencies. Iterating the local-in-time theory gives an exponential bound; Bourgain's high-low method gave the first polynomial-in-time bounds for periodic cubic NLS, and the I-method (a scale-by-scale modification of the energy by a Fourier multiplier $D$ that equals the identity below a frequency scale $N$ and $|\nabla|$ above it, introduced by Colliander–Keel–Staffilani–Takaoka–Tao) turned this into a flexible energy-increment iteration; Bourgain also proved polynomial bounds for related equations on $\mathbb{R}^d$. Strichartz estimates are space-time bounds for the free Schrödinger flow that underpin the nonlinear theory; on irrational tori their strength depends on the arithmetic of $\ell$, via how well the lattice values of the quadratic form $n_1/\ell_1^2+n_2/\ell_2^2+n_3/\ell_3^2$ avoid zero — a typical Diophantine (small-divisor) condition. The $\ell^2$ decoupling theorem of Bourgain and Demeter (Ann. of Math. 2015) yielded optimal Strichartz estimates on irrational tori, and Deng, Germain, and Guth proved long-time Strichartz estimates on irrational tori that improve on rational ones over long intervals. Deng and Germain then combined these with the upside-down I-method to control nonlinear growth: for $3<p<5$ and data with $\|u(0)\|_{H^2}=A$ and energy $E$, (1) for every $\ell$, $\|u(t)\|_{H^2}\lesssim A+(1+|t|)^{2/(5-p)+\delta}$ for any $\delta>0$, and (2) for generic $\ell$ (excluding a measure-zero subset of $(\mathbb{R}^+)^3$), $\|u(t)\|_{H^2}\lesssim A+(1+|t|)^{2/(5-p+\theta(p))}$ with $\theta(p)=\frac{\min(p-3,5-p)}{182}>0$ — a strictly better exponent thanks to irrationality. The constants depend on $\ell$, $E$, $p$ (and $\delta$) but not on $A$ or $t$. The authors state that this $\theta(p)$ is 'clearly far from optimal', that the result extends to $1<s\le 2$ and to $s>2$ only if the nonlinearity is regularized (the map $u\mapsto|u|^{p-1}u$ is not smooth), and that the improved bound is proved for generic $\ell$ and under some explicit sufficient Diophantine conditions on $\ell$, without any characterization of the sharp arithmetic hypotheses. The endpoint $p=5$ (energy-critical in three dimensions, i.e. the scaling that leaves the energy invariant) is outside their theorem; the first author treated it separately only for small energy. In parallel, the constructive side of weak turbulence has advanced in adjacent settings: Giuliani and Guardia constructed solutions of the two-dimensional cubic NLS on irrational tori whose Sobolev norms grow by any prescribed factor, showing that growth on irrational tori genuinely occurs, but in a different dimension and power than the $3<p<5$ upper-bound theory.

Problem Statement

For the defocusing nonlinear Schrödinger equation

$$(i\partial_t+\Delta)u=|u|^{p-1}u,\qquad 3<p\le 5,$$

on the rectangular torus $\mathbb{T}^3_\ell=[0,\ell_1]\times[0,\ell_2]\times[0,\ell_3]$ with conserved energy $E_\ell[u]$, take data $u(0)\in H^2$ with $\|u(0)\|_{H^2}=A$. The benchmark is the Deng–Germain theorem: for $3<p<5$, $\|u(t)\|_{H^2}\lesssim A+(1+|t|)^{2/(5-p)+\delta}$ for any $\ell$ and any $\delta>0$, and, for generic $\ell$ (the complement of a measure-zero subset of $(\mathbb{R}^+)^3$), $\|u(t)\|_{H^2}\lesssim A+(1+|t|)^{2/(5-p+\theta(p))}$ with $\theta(p)=\frac{\min(p-3,5-p)}{182}$. The problem is to sharpen and complete this result along the directions left open by this theorem and its accompanying remarks, with the common objective of determining the sharp rate of long-time Sobolev-norm growth and its dependence on the arithmetic of the torus:

1. Improve the irrationality gain: prove, for the unmodified equation with $3<p<5$, a polynomial growth bound $\|u(t)\|_{H^2}\lesssim A+(1+|t|)^{2/(5-p+\theta)}$ with a strictly larger admissible $\theta>\theta(p)$ (uniformly in $t$, with constants independent of $A$ and $t$), and in the strongest form determine the optimal growth exponent for generic or explicitly Diophantine $\ell$, ideally accompanied by constructions showing the exponent cannot be improved.
2. Characterize the arithmetic hypotheses: identify necessary and sufficient Diophantine-type conditions on the side lengths $\ell$ under which the improved bound holds, including the optimal exponents in those conditions — in particular, whether the improvement fails (reverting to the $2/(5-p)+\delta$ rate) on tori violating such conditions.
3. Extend the regularity range: establish the improved polynomial bounds for Sobolev indices $s>2$ for the genuine (unregularized) nonlinearity $|u|^{p-1}u$, beyond the essentially settled range $1<s\le 2$.
4. Treat the energy-critical endpoint $p=5$ at arbitrary (not necessarily small) energy: prove polynomial-in-time bounds on $\|u(t)\|_{H^s}$ for global solutions on generic irrational tori, removing the small-energy hypothesis of Deng's companion work.

The supercritical regime $p>5$ is deliberately outside this problem window: global well-posedness itself is unknown there, so any growth-control statement is blocked by a logically prior and different question. The four linked components are posed independently of one another and concern the same equation, torus class, norms, and uniformity conventions as the benchmark theorem.

References

1. Yu Deng, Pierre Germain, 'Growth of Solutions to NLS on Irrational Tori', International Mathematics Research Notices 2019(9), 2919–2950 (online 2017), DOI 10.1093/imrn/rnx210, arXiv:1702.04978, https://arxiv.org/abs/1702.04978
2. Yu Deng, 'On Growth of Sobolev Norms for Energy Critical NLS on Irrational Tori: Small Energy Case', Communications on Pure and Applied Mathematics 72 (2019), no. 4, 801–834 (online 2018), DOI 10.1002/cpa.21797, arXiv:1702.05617, https://arxiv.org/abs/1702.05617
3. Yu Deng, Pierre Germain, Larry Guth, 'Strichartz estimates for the Schrödinger equation on irrational tori', Journal of Functional Analysis 273 (2017), no. 9, 2846–2869, DOI 10.1016/j.jfa.2017.05.011, https://doi.org/10.1016/j.jfa.2017.05.011
4. Jean Bourgain, 'On growth in time of Sobolev norms of smooth solutions of nonlinear Schrödinger equations in $\mathbb{R}^D$', Journal d'Analyse Mathématique 72 (1997), 299–310, DOI 10.1007/bf02843163, https://doi.org/10.1007/bf02843163
5. Jean Bourgain, Ciprian Demeter, 'The proof of the $\ell^2$ decoupling conjecture', Annals of Mathematics (2) 182 (2015), no. 1, 351–389, DOI 10.4007/annals.2015.182.1.9, https://doi.org/10.4007/annals.2015.182.1.9
6. Zihua Guo, Tadahiro Oh, Yuzhao Wang, 'Strichartz estimates for Schrödinger equations on irrational tori', Proceedings of the London Mathematical Society 109 (2014), no. 4, 975–1013, DOI 10.1112/plms/pdu025, arXiv:1306.4973, https://doi.org/10.1112/plms/pdu025
7. Chenjie Fan, Gigliola Staffilani, Hong Wang, Bobby Wilson, 'On a bilinear Strichartz estimate on irrational tori', Analysis & PDE 11 (2018), no. 4, 919–944, DOI 10.2140/apde.2018.11.919, arXiv:1612.08460, https://arxiv.org/abs/1612.08460
8. Gigliola Staffilani, Bobby Wilson, 'Stability of the Cubic Nonlinear Schrödinger Equation on an Irrational Torus', SIAM Journal on Mathematical Analysis 52 (2020), no. 2, 1318–1342, DOI 10.1137/18m1179195, https://doi.org/10.1137/18m1179195
9. Filippo Giuliani, Marcel Guardia, 'Sobolev norms explosion for the cubic NLS on irrational tori', Nonlinear Analysis 220 (2022), 112865, DOI 10.1016/j.na.2022.112865, arXiv:2110.15845, https://arxiv.org/abs/2110.15845
10. Alexander Hrabski, Yulin Pan, Gigliola Staffilani, Bobby Wilson, 'Energy transfer for solutions to the nonlinear Schrödinger equation on irrational tori', arXiv:2107.01459, https://arxiv.org/abs/2107.01459
11. Roberto Feola, Jessica Elisa Massetti, 'On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori', Journal of Functional Analysis 287 (2024), no. 12, 110648, DOI 10.1016/j.jfa.2024.110648, arXiv:2303.07459, https://arxiv.org/abs/2303.07459
12. Joackim Bernier, Nicolas Camps, 'Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori', Journal de l'École polytechnique — Mathématiques 12 (2025), 713–800, DOI 10.5802/jep.300, arXiv:2402.04122, https://doi.org/10.5802/jep.300
13. Yangkendi Deng, Boning Di, Jiao Ma, Dunyan Yan, Kailong Yang, 'Long-time Strichartz estimates on 3D waveguide with applications', arXiv:2601.20392 (2026), https://arxiv.org/abs/2601.20392