Global Well-Posedness near Rayleigh-Jeans Equilibria for the Cubic NLS Wave Kinetic Equation

Abstract: We study the dynamics of the kinetic wave equation associated to the three dimensional Schrödinger equation close to Rayleigh-Jeans equilibria. We first prove that the linearised operator generates a semigroup of contractions in $L^{2((0,\infty);\sqrt} ω\ddω)$. Considering the family of nonsingular Rayleigh-Jeans spectra, we prove that the linearised operator possesses a spectral gap, despite the non-compactness of the integral collisional operator, and thus obtain an exponential relaxation for the linear semigroup. We then prove bilinear and trilinear estimates in the relevant norm for the nonlinear terms and deduce global well-posedness and exponential relaxation for sufficiently small relative perturbations of Rayleigh-Jeans equilibria. To our knowledge, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation.

• The paper proves global strong well-posedness and exponential relaxation for sufficiently small perturbations of every nonsingular Rayleigh–Jeans equilibrium with μ>0 in weighted L².
• The authors establish a spectral gap for the linearised collision operator despite its non-compact integral component by combining low- and high-frequency coercivity with local compactness.
• The results identify a perturbative regime where condensation does not occur, while highlighting open questions near the singular μ=0 equilibrium and for larger perturbations.

This paper establishes the nonlinear asymptotic stability of nonsingular Rayleigh–Jeans (RJ) equilibria for the spatially homogeneous, isotropic four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation on $\mathbb{R}^3$, working on the whole frequency domain without ultraviolet cutoff. The authors prove that the linearised collision operator around each nonsingular RJ spectrum generates a contraction semigroup in $L^2((0,\infty);\sqrt{\omega}\,\omega)$ with a spectral gap—despite the non-compactness of its integral part—and combine this with bilinear and trilinear bounds to obtain global well-posedness and exponential relaxation for sufficiently small perturbations. According to the authors, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous wave kinetic equation of cubic NLS.

Setting and main results

The isotropic kinetic equation under study is

$\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$

with dispersion $\omega(k)=|k|^2$, where the collision operator involves the resonant kernel $\Omega(\omega,\omega_1,\omega_2,\omega_3)$ over the domain $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$. The formal detailed-balance equilibria are the RJ spectra $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$; after absorbing $T$ into time, the nonsingular family is $n_\mu(\omega)=(\omega+\mu)^{-1}$ with $\mu>0$. Writing solutions as $L^2((0,\infty);\sqrt{\omega}\,\omega)$0 yields a perturbation equation

$L^2((0,\infty);\sqrt{\omega}\,\omega)$1

where $L^2((0,\infty);\sqrt{\omega}\,\omega)$2 splits into a multiplication part $L^2((0,\infty);\sqrt{\omega}\,\omega)$3 and an integral part $L^2((0,\infty);\sqrt{\omega}\,\omega)$4, and $L^2((0,\infty);\sqrt{\omega}\,\omega)$5, $L^2((0,\infty);\sqrt{\omega}\,\omega)$6 collect quadratic and cubic nonlinearities.

The functional space is $L^2((0,\infty);\sqrt{\omega}\,\omega)$7, which up to the constant $L^2((0,\infty);\sqrt{\omega}\,\omega)$8 is the radial subspace of $L^2((0,\infty);\sqrt{\omega}\,\omega)$9; it is natural because $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$0 there (for $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$1) and because the symmetrised Dirichlet form is well defined in it. The kernel of $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$2 is characterised via collisional invariants: in frequency variables, $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$3, and the $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$4 constraint forces $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$5, so $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$6.

The main results are:

• Linear theory: $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$7 is symmetric, nonnegative, bounded, and m-accretive on $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$8 for all $\partial_t f(t,\omega) = \mathcal{C}(f)(\omega),$9; hence $\omega(k)=|k|^2$0 is a contraction semigroup with $\omega(k)=|k|^2$1 and strong convergence $\omega(k)=|k|^2$2. However, $\omega(k)=|k|^2$3 is not compact for any $\omega(k)=|k|^2$4.
• Spectral gap: for every $\omega(k)=|k|^2$5 there exists $\omega(k)=|k|^2$6 such that $\omega(k)=|k|^2$7.
• Nonlinear stability: for any $\omega(k)=|k|^2$8 there exists $\omega(k)=|k|^2$9 such that if $\Omega(\omega,\omega_1,\omega_2,\omega_3)$0 and $\Omega(\omega,\omega_1,\omega_2,\omega_3)$1, then the solution is global in $\Omega(\omega,\omega_1,\omega_2,\omega_3)$2 and satisfies $\Omega(\omega,\omega_1,\omega_2,\omega_3)$3. Positivity of $\Omega(\omega,\omega_1,\omega_2,\omega_3)$4 is propagated whenever $\Omega(\omega,\omega_1,\omega_2,\omega_3)$5.

The exponential relaxation result has a direct implication for the condensation question raised by Deng and Hani: the paper identifies a rigorous perturbative regime in which condensation does not occur—small perturbations of nonsingular RJ equilibria remain global and relax exponentially to equilibrium.

The obstruction from non-compactness

A central technical point is that the classical Grad compact-perturbation argument fails here. In the Boltzmann theory of gases and phonon kinetics, the linearised operator splits into a coercive multiplication part plus a compact integral part, so the essential spectrum is unaffected by the latter. Here, the authors prove that $\Omega(\omega,\omega_1,\omega_2,\omega_3)$6 is not compact on $\Omega(\omega,\omega_1,\omega_2,\omega_3)$7 for any $\Omega(\omega,\omega_1,\omega_2,\omega_3)$8, via a dilation argument: conjugating $\Omega(\omega,\omega_1,\omega_2,\omega_3)$9 by the unitary $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$0 gives $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$1, and a contradiction argument using monotone convergence shows that high-frequency dilates of a fixed compactly supported function violate the strong-to-zero property required of compact operators. Comparing with the box-truncated problem of Menegaki [2306.xxxx context], where compactness holds, the failure is attributable precisely to the unbounded high-frequency domain. Consequently, the spectral gap must be obtained by other means.

Spectral gap without compactness

The proof of the Poincaré inequality proceeds by contradiction, assuming a normalised sequence $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$2 with Dirichlet form $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$3, and showing that no mass can escape to either end of the frequency axis:

• Coercivity at low frequencies: using the explicit lower bound $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$4 for the multiplier of $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$5, together with an $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$6 bound on $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$7, one obtains $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$8.
• Local compactness: $$D(\omega)=\{(\omega_1,\omega_2):\ \omega_1+\omega_2\ge\omega\}$$9 is Hilbert–Schmidt, hence compact.
• Coercivity at high frequencies — the delicate part: restricting the Dirichlet form to a rectangle $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$0 of comparable frequencies and passing to logarithmic variables $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$1 with $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$2, Plancherel reduces the restricted Dirichlet form to a Fourier multiplier $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$3. Pointwise positivity follows because the vanishing of the integrand would force $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$4, impossible for real $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$5; uniform positivity as $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$6 follows by non-stationary phase, since all mixed oscillatory terms decay like $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$7. A localisation argument with a carefully rescaled cutoff $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$8, $$n_{T,\mu_{\mathrm{chem}}}(\omega)=T/(\omega-\mu_{\mathrm{chem}})$$9, transfers this coercivity to arbitrary functions supported near infinity, yielding $T$0 with $T$1. Notably, the standard cutoff ($T$2) would fail to give a uniform-in-$T$3 bound.

Combining these ingredients shows $T$4, contradicting the lower bound $T$5 derived from coercivity of $T$6. This establishes the spectral gap and, by a Grönwall energy estimate, exponential decay $T$7.

Nonlinear estimates and global well-posedness

The bilinear and trilinear bounds

$T$8

are proved by dyadic decomposition of the frequency axis. Terms where $T$9 factors out are handled directly by Hölder; the delicate term $n_\mu(\omega)=(\omega+\mu)^{-1}$0 requires rewriting the resonance relation as a convolution on $n_\mu(\omega)=(\omega+\mu)^{-1}$1 (via reflection of the lowest-frequency factor) and applying Young's inequality with the $n_\mu(\omega)=(\omega+\mu)^{-1}$2 norm placed on the highest-frequency input. Resonance constraints force either $n_\mu(\omega)=(\omega+\mu)^{-1}$3 or $n_\mu(\omega)=(\omega+\mu)^{-1}$4 across dyadic scales, and geometric summability closes the estimate. A scaling analysis shows the constants blow up as $n_\mu(\omega)=(\omega+\mu)^{-1}$5: specifically $n_\mu(\omega)=(\omega+\mu)^{-1}$6 for the quadratic term and $n_\mu(\omega)=(\omega+\mu)^{-1}$7 for the cubic term, so the restriction $n_\mu(\omega)=(\omega+\mu)^{-1}$8 is structural rather than merely technical.

Local well-posedness follows by Banach fixed point, with the standard blow-up alternative. Global existence for small data uses conservation of collision invariants to propagate orthogonality to $n_\mu(\omega)=(\omega+\mu)^{-1}$9, then the differential inequality

$\mu>0$0

which yields exponential decay at rate $\mu>0$1 provided $\mu>0$2; continuity prevents the norm from ever reaching $\mu>0$3, and the blow-up criterion extends the solution globally. Propagation of nonnegativity is proved via a truncated-operator approximation and a Grönwall argument on the negative part of $\mu>0$4.

Relation to prior work and limitations

Several features delimit the scope of the results. First, the stability theorem requires $\mu>0$5; the singular case $\mu>0$6 is excluded, and in the authors' companion work the singular RJ profile exhibits finite-time condensation (formation of a Dirac mass at zero) for truncated RJ initial data. The two behaviours are reconciled by topology: the weight $\mu>0$7 vanishes at the origin, so condensation at zero may be invisible in $\mu>0$8. Second, the perturbation size $\mu>0$9 depends on the spectral gap and shrinks as $L^2((0,\infty);\sqrt{\omega}\,\omega)$00 through the constants $L^2((0,\infty);\sqrt{\omega}\,\omega)$01; whether stability persists for larger perturbations or up to the singular limit is open. Third, the result concerns the spatially homogeneous, isotropic equation only; the inhomogeneous problem, where transport provides additional dispersion, is treated by different methods in the literature. Fourth, existing local well-posedness theories in nearly critical weighted spaces require faster-than-RJ decay at infinity, so they do not cover RJ neighbourhoods—the present paper supplies its own local theory instead. Finally, the question posed by Deng and Hani of whether condensation occurs generically or only for particular solution classes remains open outside the perturbative regime identified here.

Conclusion

The paper resolves a long-standing open problem in wave kinetic theory by proving that nonsingular Rayleigh–Jeans equilibria of the homogeneous four-wave kinetic equation for cubic NLS are exponentially asymptotically stable in their natural weighted $L^2((0,\infty);\sqrt{\omega}\,\omega)$02 topology, on the whole frequency domain and without ultraviolet cutoff. The main methodological contribution is a spectral gap proof that circumvents the failure of compactness of the integral part of the linearised operator, combining low-frequency tightness, local Hilbert–Schmidt compactness, and a Fourier-analytic coercivity estimate at infinity transferred through a logarithmically rescaled cutoff. Together with scale-sharp multilinear estimates, this yields the first global strong well-posedness and relaxation result near a nonzero thermodynamic equilibrium for this equation, and delineates a regime in which Bose-type condensation does not occur.