Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the $3d$ ball

Abstract: We construct probabilistic strong solutions to the cubic Schrödinger equation on the three-dimensional ball with radial initial data, which is a significant improvement of a result by Bourgain--Bulut. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng--Nahmod--Yue. We achieve this result through gauge transformations that do not modify the equation, combined with a refined modulation analysis using random averaging operators.

• The paper constructs unique probabilistic strong solutions for radial cubic NLS on the 3D ball when Gaussian initial data satisfy α>15/16, improving the previous threshold α=1.
• The authors linearly twist the dispersion relation to neutralize frequency-dependent resonances and convert the main cubic obstruction into a manageable non-resonant quintic term.
• A random averaging operator ansatz removes singular high–low interactions, enabling frequency-truncated solutions with the full nonlinearity to converge locally in time below the critical Sobolev regularity s=1/2.

Overview and main result

The paper by Burq, Camps, Sun, and Tzvetkov studies the defocusing cubic Schrödinger equation $(i\partial_t+\Delta)u=|u|^2u$ with Dirichlet boundary conditions on the three-dimensional ball $\mathbf{B}$ of radius $\pi$, restricted to radial data. The authors construct probabilistic strong solutions for Gaussian random initial data

$$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$

where $\mathbf{e}_n$ are the radial eigenfunctions of the Dirichlet Laplacian and $(g_n)$ are independent standard complex Gaussians. The main theorem establishes that for $\alpha>15/16$, there exists a set of full $\mu_\alpha$-measure on which the frequency-truncated solutions $u_N$ (evolved by the full cubic nonlinearity, not the Galerkin-truncated one) converge locally in time in $$L^\infty([-T,T];H_{\mathrm{rad}}^{\alpha-\frac12-}(\mathbf{B}))$$ to a unique distributional solution.

This result is a substantial improvement of the theorem of Bourgain–Bulut, which required $\mathbf{B}$0 — precisely the regularity of typical functions in the support of the Gibbs measure — and which relied on the Galerkin projection $\mathbf{B}$1 on the right-hand side. The new result reaches $\mathbf{B}$2, i.e., Sobolev regularities $\mathbf{B}$3, strictly below the critical Sobolev regularity $\mathbf{B}$4 of the equation. Moreover, since the probabilistic scaling threshold of Deng–Nahmod–Yue coincides with $\mathbf{B}$5, the constructed flow is supercritical with respect to probabilistic scaling. The result applies equally to both the Galerkin-truncated and the full-nonlinear approximations, which the Bourgain–Bulut argument cannot accommodate.

The obstruction: frequency-dependent resonance and the failure of global gauges

The proof of Bourgain–Bulut proceeds by controlling the distance $\mathbf{B}$6 through an estimate of the form

$\mathbf{B}$7

whose logarithmic divergence arises from the purely high-frequency resonant interaction

$\mathbf{B}$8

where $\mathbf{B}$9 is the four-eigenfunction correlation coefficient. On the torus, analogous non-smoothing terms (such as $\pi$0 in Wick-ordered models) can be removed by a global gauge transformation. Here this is impossible: the mode equation takes the form

$\pi$1

with a divergent, frequency-dependent, real-valued coefficient $\pi$2; no single global gauge removes all such contributions while preserving the structure of the nonlinearity. This is a structural obstruction specific to the non-translation-invariant geometry of the ball.

The key algebraic observation is that $\pi$3 has favorable structure permitting a linear correction of the dispersion relation rather than a nonlinear gauge transform. Defining

$\pi$4

one has $\pi$5 for $\pi$6 (and $\pi$7 for $\pi$8). The authors then rewrite the resonant term as $\pi$9 plus a centered random term, exploiting the identity $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$0, in which the resonant contributions cancel because they are real-valued. The critical cubic interaction is thereby converted into a non-resonant quintic term $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$1, amenable to perturbative treatment. This linear frequency-twisting strategy is in sharp contrast with the nonlinear gauge transforms of Oh–Tzvetkov–Wang, and is inspired by Tao's gauges for wave maps and Benjamin–Ono.

Random averaging operator ansatz

The second structural ingredient is a random averaging operator (RAO) ansatz, a frequency-localized variant of the paracontrolled calculus of Gubinelli–Imkeller–Perkowski, following Deng–Nahmod–Yue and the sphere-based framework of the authors' earlier work. The high-frequency increment $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$2 is decomposed as $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$3, where the colored Gaussian term $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$4 solves, mode by mode, the linear ODE

$$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$5

so that the RAO reduces, in this radial one-dimensional setting, to multiplication by the unit-modulus phase

$$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$6

Crucially, $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$7 is $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$8-measurable, hence independent of the Gaussian variables $$u|_{t=0}=\phi_\alpha(\omega,x)=\sum_{n\geq1}\frac{g_n(\omega)}{n^\alpha}\,\mathbf{e}_n(x),\qquad \alpha\leq1,$$9, and the singular high$\mathbf{e}_n$0low$\mathbf{e}_n$1low contribution $\mathbf{e}_n$2 is exactly subtracted in the equation for the smoother remainder $\mathbf{e}_n$3. This yields the Gronwall-friendly estimate

$\mathbf{e}_n$4

without the logarithmic divergence, whereas the Bourgain–Bulut approach in this regime produces a divergent factor $\mathbf{e}_n$5 with $\mathbf{e}_n$6 as $\mathbf{e}_n$7.

The rigorous implementation uses a truncated Duhamel operator and an inductive scheme over dyadic scales, propagating a quantitative property $\mathbf{e}_n$8 comprising: (i) bounds on the modified phases $\mathbf{e}_n$9 in twisted Fourier–Lebesgue norms $(g_n)$0; (ii) a hypercontractive bound on centered quadratic sums of the Gaussians weighted by $(g_n)$1; and (iii) $(g_n)$2 bounds on $(g_n)$3 with off-support decay. The key induction step holds $(g_n)$4-certainly, and iteration over dyadic scales gives the quantitative main theorem: with $(g_n)$5, the approximants converge on $(g_n)$6 outside a set of $(g_n)$7-measure at most $(g_n)$8, and the limit decomposes as $(g_n)$9 with $\alpha>15/16$0 for some $\alpha>15/16$1 — i.e., the remainder is smoother than the linear evolution, isolating all roughness in the explicitly constructed colored Gaussian component.

Counting, Strichartz, and probabilistic tools

The deterministic core consists of new counting estimates for multi-indices restricted to thickened level sets of the resonance function $\alpha>15/16$2. The divisor bound with perturbed frequencies gives

$\alpha>15/16$3

with an $\alpha>15/16$4 bound when $\alpha>15/16$5 — a gain attributable to the sublinear growth of $\alpha>15/16$6. These yield Strichartz-type multilinear estimates in frequency space, including a variant with an $\alpha>15/16$7 input on one frequency and an almost-orthogonality mechanism for the high-high-very-low regime. On the probabilistic side, conditional Wiener chaos estimates combined with the non-commutative Khintchine inequality (following Bringmann, and Kaneko's recent formulation) produce moment bounds for fully non-resonant trilinear interactions of colored Gaussian inputs, with a square-root gain over deterministic counting. A technical cost is a $\alpha>15/16$8 loss in the non-commutative Khintchine inequality when the two random scales coincide or are widely separated; the former is resolved by decoupling with an independent Gaussian copy, the latter by a modulation gain via the partially resonant analysis.

Multilinear estimates and the threshold $\alpha>15/16$9

The trilinear estimates split into fully non-resonant interactions $\mu_\alpha$0, partially resonant ones $\mu_\alpha$1 (which require a refined modulation analysis in the spirit of the authors' $\mu_\alpha$2 work, but now purely in frequency space, and without a cubic Wick renormalization — an effect of the non-translation-invariant geometry), and quadratic/operator bounds. The quintic estimates reduce, via a Hölder–Young convolution lemma, to pointwise-in-time bounds on the quartic form $\mu_\alpha$3, proved deterministically when at least two inputs are of type (D) and probabilistically when at least three inputs are of type (C), with the uniform gain exponent

$\mu_\alpha$4

which, combined with the requirement that the range $\mu_\alpha$5 be non-empty, forces $\mu_\alpha$6. The authors state plainly that this threshold is not optimal: heuristically, after removing the critical resonant term, the non-resonant high$\mu_\alpha$7high typical regularity is $\mu_\alpha$8, suggesting $\mu_\alpha$9 as the natural limit of the method; the gap is due to a non-optimal factor $u_N$0 lost in the counting estimates associated with the modified dispersion relation, plus a technical constraint from the high–high interaction in the quintic term handled only deterministically.

Limitations and open questions

Several restrictions are conceded explicitly. The constraint $u_N$1 is an artifact of the present counting estimates and the purely deterministic treatment of one quintic configuration; closing the gap to the conjectured threshold $u_N$2 is left open. The case $u_N$3 is out of reach: on $u_N$4 (or $u_N$5) the cubic NLS is energy-critical and the Strichartz estimates on the sphere suffer a derivative loss, in contrast with $u_N$6 where arithmetic structure yields stronger estimates and Yue constructed a global flow in the energy space. The result is local in time; global extension for $u_N$7 is not addressed, although the authors note that at $u_N$8 the invariant measure argument of Deng–Nahmod–Yue should extend the structural decomposition globally. Finally, the analysis is expected to transfer to zonal spherical harmonics on $u_N$9 — technically simpler since boundary effects, controlled here by the nontrivial decay estimate $$L^\infty([-T,T];H_{\mathrm{rad}}^{\alpha-\frac12-}(\mathbf{B}))$$0 for $$L^\infty([-T,T];H_{\mathrm{rad}}^{\alpha-\frac12-}(\mathbf{B}))$$1, are absent — but this extension is not carried out.

Conclusion

The paper resolves the main technical obstruction left open by Bourgain–Bulut for the Gibbs-measure problem for the radial cubic NLS on the $$L^\infty([-T,T];H_{\mathrm{rad}}^{\alpha-\frac12-}(\mathbf{B}))$$2 ball: a frequency-dependent resonant coefficient that no global gauge can remove. By twisting the dispersion relation linearly — converting the critical cubic resonance into a perturbative non-resonant quintic term — and by implementing a random averaging operator ansatz in which the averaging operator degenerates to a random phase, the authors obtain a Gronwall-compatible convergence scheme and construct probabilistic strong solutions at regularities strictly below the Gibbs-measure threshold, in a regime supercritical for probabilistic scaling. The price is a quantitative threshold $$L^\infty([-T,T];H_{\mathrm{rad}}^{\alpha-\frac12-}(\mathbf{B}))$$3, conjecturally improvable to $$L^\infty([-T,T];H_{\mathrm{rad}}^{\alpha-\frac12-}(\mathbf{B}))$$4 within the same framework, and the restriction to radial data on the ball (with a plausible extension to zonal modes on $$L^\infty([-T,T];H_{\mathrm{rad}}^{\alpha-\frac12-}(\mathbf{B}))$$5).