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Condensation and Non-condensation in the Quasilinear 4-wave Kinetic Equation

Siwei Luo · Jian-Guo Liu

  • Preprint
  • 2026
  • arXiv:2610.25207
  • math.AP · math-ph
  • 52 pages · 1 figure · 1 table
  • 6 views
Two panels showing enlargement of a positive-frequency interval and production of output frequencies near zero.
Figure 1 Frequency regions used to enlarge a positive-density interval and produce output frequencies near zero.

Abstract

We study finite-time condensation at zero frequency for the isotropic quasilinear 4-wave kinetic equation in three dimensions. With Schrödinger dispersion and a product interaction coefficient with exponent \(\beta\in (0,3/4]\), we construct global measure solutions \((g_t)_{t\geq 0}\) for every nonnegative initial measure \(g_0\) with finite wave action and energy. We then establish the sharp threshold \(\beta=1/4\) for finite-time condensation of the constructed solutions. For \(\beta\in [1/4,3/4],\) the constructed solutions satisfy \(g_t(\{0\})=g_0(\{0\})\) for any \(t\geq 0\). For \(\beta\in (0,1/4)\), every constructed solution with \((g_0)_{\rm ac}\neq 0\) satisfies \[\|g_t-\mathcal N_0\delta_0\|_{\mathrm{TV}} \le C\mathcal N_0(1+\mathcal N_0^2t)^{-(1-4\beta)/40}\] for all \(t\ge0\), where \(\mathcal N_0=g_0([0,\infty))\) and \(C=C(\beta,g_0/\mathcal N_0)>0\). In particular, for every \(\zeta\in(0,1)\), there exists \(T_\zeta<\infty\) such that \(g_t(\{0\})\ge(1-\zeta)\mathcal N_0\) for all \(t\ge T_\zeta\).

Introduction

The wave kinetic equation (WKE) describes the evolution of the wave spectrum in the weak-turbulence approximation to the cubic nonlinear Schrödinger equation [DNPZ92]. Resonant interactions can transfer wave action towards low frequencies and energy towards high frequencies, giving rise to an inverse wave-action cascade and a direct energy cascade [ZLF25, Naz11]. The accumulation of a positive amount of wave action at zero frequency can form a condensate, while at the other end of the spectrum, the transfer of a positive amount of energy to infinite frequency leads to possible energy loss from the kinetic solution in finite time.

Wave kinetic equations can also describe more general Hamiltonian wave systems [ZLF25, MMT97]. In these systems, the coupling strength can depend on the participating wave vectors through an interaction coefficient. In this paper, we study how this dependence affects finite-time condensate formation. We consider a family of 4-wave kinetic equations in three dimensions with Schrödinger dispersion \(\omega( k)=| k|^2\) and a product interaction coefficient depending on an exponent \(\beta \in (0,3/4]\). Varying \(\beta\) can change the interaction strength for different frequency scales and lead to possible phase transition of action condensation. Indeed, we construct global isotropic measure solutions and then prove a sharp threshold in \(\beta\) for finite-time condensate formation for the constructed solutions.

The model

We consider the three-dimensional quasilinear Schrödinger equation \[i\partial_tu+\Delta u =\kappa|\nabla|^\beta \left(\bigl||\nabla|^\beta u\bigr|^2|\nabla|^\beta u\right), \qquad x\in\mathbb R^3,\tag{1.1}\] where \(\kappa\in\mathbb R\setminus\{0\}\), \(\beta\in(0,3/4]\), and \(|\nabla|^\beta=(-\Delta)^{\beta/2}\). It is generated by the Hamiltonian \[\mathcal H(u) =\int_{\mathbb R^3}|\nabla u|^2\,d x +\frac{\kappa}{2}\int_{\mathbb R^3} \bigl||\nabla|^\beta u\bigr|^4\,d x,\tag{1.2}\] through \(i\partial_tu=\delta\mathcal H/\delta\overline u\). At \(\beta=0\), this equation reduces to the cubic nonlinear Schrödinger equation.

Now consider the corresponding wave kinetic equation of (1.1). For a spatially homogeneous random wave field, let \(n(t, k)\) denote its wave-action spectrum. The standard weak-turbulence theory formally leads to a 4-wave kinetic equation [ZLF25, Naz11]. After absorbing the prefactor proportional to \(\kappa^2\) into time, we obtain \[\partial_tn_4 =\mathcal C_\beta[n]( k_4),\qquad n_j=n(t, k_j),\quad \omega_j=\omega(k_j)=|k_j|^2,\quad j=1,2,3,4.\tag{1.3}\] \[\mathcal C_\beta[n]( k_4) ={}\int_{\mathbb R^9} |T_{1234}|^2 \delta^{(3)}(\Sigma)\delta(\Omega) \bigl[n_1n_2(n_3+n_4)-n_3n_4(n_1+n_2)\bigr] \,d k_1\,d k_2\,d k_3.\tag{1.4}\] Here \(\Sigma= k_1+ k_2 - k_3- k_4\) and \(\Omega=\omega_1+\omega_2-\omega_3-\omega_4\). Since \(|\nabla|^\beta\) has Fourier symbol \(| k|^\beta\), the quartic Hamiltonian (1.2) gives the interaction coefficient \(T_{1234}=(| k_1|| k_2| | k_3|| k_4|)^\beta\). The Dirac delta functions impose the resonance relations \(\Sigma=0\) and \(\Omega=0\) in (1.3).

The periodic version of (1.1) with \(\kappa=1\) and its associated kinetic equation are discussed by Ampatzoglou and Léger [AL25]. Zhu [Zhu26] then considers the same product interaction coefficient with power-law dispersion \(\omega( k)=| k|^a\), \(1\le a\le5\). The derivative nonlinearity also appears in the one-dimensional family introduced by Majda, McLaughlin, and Tabak [MMT97] to study dispersive wave turbulence, allowing both the dispersion relation and the interaction coefficient to vary. Here we fix the Schrödinger dispersion and study how the interaction exponent \(\beta\) affects condensate formation.

Define the total wave action and the total energy of the kinetic equation \[\mathcal N(t)=\int_{\mathbb R^3} n(t, k)\,d k,\qquad \mathcal E(t)=\int_{\mathbb R^3} | k|^2n(t, k)\,d k.\tag{1.5}\] For isotropic spectra, we write \(n(t,\omega)\) for \(n(t,k)\), where \(\omega=|k|^2\). The change of variable gives the wave-action measure \(g_t(d\omega)=g(t,\omega)\,d\omega\), with \(g(t,\omega)=2\pi\sqrt{\omega}\,n(t,\omega)\). Thus \[\mathcal N(t)=\int_{[0,\infty)}dg_t,\qquad \mathcal E(t)=\int_{[0,\infty)}\omega\,dg_t.\tag{1.6}\] Note that we allow \(g_t\) to be a measure on \([0,\infty)\) instead of a function to include possible condensates. We now state the radial equation in weak form of (1.3) with respect to \(g_t\).

Throughout the paper, we write \(a\wedge b:=\min\{a,b\}\) and \(a\vee b:=\max\{a,b\}\).

Set \(\omega_4=\omega_1+\omega_2-\omega_3\). For \(\omega_1,\omega_2,\omega_3>0\), define \[K_\beta(\omega_1,\omega_2,\omega_3) =(\omega_1\omega_2\omega_3\omega_4)^\beta \dfrac{\min\{\sqrt{\omega_1},\sqrt{\omega_2},\sqrt{\omega_3},\sqrt{\omega_4}\}} {\sqrt{\omega_1\omega_2\omega_3}}\tag{1.7}\] when \(\omega_4>0\). We set \(K_\beta=0\) whenever one of the four frequencies is nonpositive.

Let \(\mathcal M_1\) be the Banach space of finite signed Borel measures \(g\) on \([0,\infty)\) satisfying \[\|g\|_{\mathcal M_1}:=\int_{[0,\infty)}(1+\omega)\,d|g|<\infty.\tag{1.8}\] We also write \[\operatorname{Lip}_c([0,\infty)) =\{\phi\text{ Lipschitz on }[0,\infty): \phi\text{ has compact support}\}.\tag{1.9}\]

Definition 1.1 (Isotropic measure solution). Let \(g_0\in\mathcal M_1\) be nonnegative. A narrowly continuous curve \((g_t)_{t\geq0}\) of nonnegative measures in \(\mathcal M_1\) is called an isotropic measure solution of (1.3) if \(g_{t=0}=g_0\), \(\sup_{0\leq t\leq T}\bigl(\mathcal N(t)+\mathcal E(t)\bigr)<\infty\) for every \(T<\infty\), and for every \(\phi\in \operatorname{Lip_c}([0,\infty))\) and \(T<\infty\), \[\int_0^T\iiint_{[0,\infty)^3} K_\beta(\omega_1,\omega_2,\omega_3)|\Delta\phi|\, dg_\tau^{\otimes 3}\,d\tau<\infty,\tag{1.10}\] and for \(0\leq s\leq t<\infty\), \[\int_{[0,\infty)}\phi\, dg_t-\int_{[0,\infty)}\phi\, dg_s=\dfrac{1}{4}\int_s^t\iiint_{[0,\infty)^3} K_\beta(\omega_1,\omega_2,\omega_3)\Delta\phi(\omega_1,\omega_2,\omega_3)\, dg_\tau^{\otimes 3}d\tau,\tag{1.11}\] where \(dg_\tau^{\otimes 3}=dg_\tau(\omega_1)dg_\tau(\omega_2)dg_\tau(\omega_3)\) and \(\Delta\phi=\phi(\omega_3)+\phi(\omega_4)-\phi(\omega_1)-\phi(\omega_2)\) when \(\omega_4>0\). We set \(\Delta\phi=0\) when \(\omega_4\le0\).

For \(\beta=0\), the corresponding weak formulation with a non-interacting condensate is given in [EV15b], and a derivation with general interaction coefficients is given in [MA16]. For the product coefficient, one can refer to [AL25] for the details.

We now define finite-time condensation. A condensate is represented by a positive atom of \(g_t\) at \(\omega=0\), i.e. \(g_t(\{0\})>0\) for some \(t<\infty\) [EV15b, ST26a]. We say that finite-time condensate formation occurs if \(g_0(\{0\})=0\) and \(g_T(\{0\})>0\) for some \(0<T<\infty\).

Our results identify the condensation threshold in the range \(\beta\in (0,3/4]\): condensate formation can occur precisely below \(\beta=1/4\). For \(\beta>3/4\), finite wave action and energy no longer guarantee absolute convergence of the collision integral on the right-hand side of (1.11), and this range is therefore not considered in the present paper.

Remark 1.2 (Non-interacting condensate). We use a weak formulation with a non-interacting condensate here. Since the collision kernel \(K_\beta\) vanishes whenever one of the four frequencies is zero, a condensate atom already present at zero does not contribute directly to the collision integral.

Remark 1.3 (The coefficient \(1/4\) in the weak formulation). With the normalization in (1.3) and \(g(t,\omega)=2\pi\sqrt{\omega}\,n(t,\omega)\), the radial weak formulation has the coefficient \(1/4\) in (1.11). The calculation is given in [AL25].

Remark 1.4 (The boundary \(\omega_4=0\)). The zero extension of \(\Delta\phi\) in Definition 1.1 may be discontinuous at \(\omega_4=0\). However, for the passage to the limit in the weak equation in Section 2, we use a symmetrized integrand \(F_\beta[\phi]\). By Corollary 2.4, this function is continuous on \(\mathcal D_{\mathrm{ord}}\), and \(F_{\beta,N}[\phi]\to F_\beta[\phi]\) locally uniformly there.

Main results

For a nonnegative initial measure \(g_0\in\mathcal M_1\), write \[{\mathcal N}_0=\int_{[0,\infty)} dg_0,\qquad {\mathcal E}_0=\int_{[0,\infty)}\omega\,dg_0.\tag{1.12}\] In Section 2, we will construct isotropic measure solutions in the sense of Definition 1.1 by introducing symmetric cutoff equations and passing to the limit as the cutoff is removed. Let \(g^N=(g_t^N)_{t\ge0}\) denote the cutoff solution with initial value \(g_0\).

Theorem 1.5 (Global measure solutions). Let \(\beta\in (0,3/4]\) and let \(g_0\in\mathcal M_1\) be nonnegative. There exists a global isotropic measure solution \((g_t)_{t\geq0}\) of (1.11) in the sense of Definition 1.1 with initial value \(g_0\) satisfying \[\mathcal N(t)=\mathcal N_0,\qquad \mathcal E(t)\leq \mathcal E(s)\leq \mathcal E_0, \qquad 0\leq s\leq t<\infty.\tag{1.13}\]

We now state the threshold \(\beta=1/4\) for finite-time action condensation of the solutions constructed in Theorem 1.5.

Theorem 1.6 (Condensation and non-condensation threshold). Let \(\beta\in (0,3/4]\). The following statements hold.

  • If \(\beta\in [1/4,3/4]\), then every solution constructed in Theorem 1.5 and Section 2 satisfies \(g_t(\{0\})=g_0(\{0\})\) for every \(t\ge0\). In particular, if \(g_0(\{0\})=0\), no condensation forms in finite time.

  • If \(\beta\in(0,1/4)\), let \(g_0\in\mathcal M_1\) be nonnegative and have a nonzero absolutely continuous part with respect to Lebesgue measure. For every \(0<\zeta<1\), there is a finite time \(T=T(\beta,g_0,\zeta)\) such that every solution constructed in Theorem 1.5 and Section 2 with initial measure \(g_0\) satisfies \[g_t(\{0\})\ge(1-\zeta)\mathcal N_0, \qquad t\ge T.\tag{1.14}\] Moreover, there is a constant \(C\), depending only on \(\beta\) and \(g_0/\mathcal N_0\), such that all these solutions satisfy \[\|g_t-\mathcal N_0\delta_0\|_{\mathrm{TV}} \le C\mathcal N_0(1+\mathcal N_0^2t)^{-(1-4\beta)/40}, \qquad t\ge0.\tag{1.15}\]

The phase transition results for all \(\beta\in(0,3/4]\) are summarized in Table 1.

Table 1. Condensation threshold.
Parameter \(\beta\) Condensate formation
\((0,1/4)\) Every datum with a nonzero absolutely continuous part and no atom at zero condenses in finite time.
\([1/4,3/4]\) The condensate mass is constant.

Remark 1.7 (Energy conservation and loss). The energy inequality in Theorem 1.5 allows for energy loss through transfer to infinite frequency. In a forthcoming work, we will address the threshold \(\beta=2/3\) for the solutions constructed here: energy conservation for \(0<\beta\le2/3\), and finite-time energy loss for suitable initial data when \(2/3<\beta\le3/4\).

Wave turbulence theory has its origins in the study of lattice vibrations, water waves, and random dispersive waves, including the classical works of Peierls [Pei29], Phillips [Phi60], Hasselmann [Has62], Benney and Saffman [BS66], Benney and Newell [BN69], and Zakharov [Zak65, Zak68]. It provides a kinetic description of spectral transfer in weakly nonlinear wave systems, with applications to fluid dynamics, plasmas, Bose gases, and nonlinear optics. General accounts of the theory and its physical applications can be found in the books of Zakharov, Lvov, and Falkovich [ZLF25] and Nazarenko [Naz11].

For the 4-wave kinetic equation with a constant interaction coefficient associated with the cubic nonlinear Schrödinger equation, Escobedo and Velázquez [EV15b] developed a theory of isotropic weak and mild solutions, including local and global well-posedness, finite-time condensation, and long-time asymptotics. Germain, Ionescu, and Tran [GIT20] then established local well-posedness in weighted \(L^\infty\) spaces for a general class of radial dispersion relations, and in weighted \(L^2\) spaces for Schrödinger dispersion, without assuming isotropy of the initial data. Recently, for a class of convex radial dispersion relations, Staffilani and Tran [ST26b] constructed global mild radial solutions and proved transfer of energy to high frequencies as \(t\to\infty\) under suitable assumptions on the initial data. They also established sufficient conditions for finite-time condensation and estimates on non-condensation times [ST26a].

Equations with frequency-dependent interaction coefficients have also been studied. The Majda–McLaughlin–Tabak model was introduced as a family of one-dimensional equations for studying dispersive wave turbulence [MMT97], and Germain, La, and Zhang [GLZ25] established local well-posedness for its kinetic equation. Ampatzoglou and Léger [AL25] studied the same family of spatially homogeneous 4-wave kinetic equations considered here, with quadratic dispersion and interaction weight \(|T_{1234}|^2=(\omega_1\omega_2\omega_3\omega_4)^\beta =(|k_1||k_2||k_3||k_4|)^{2\beta}\), associated with a family of quasilinear Schrödinger equations. They proved local well-posedness in suitable weighted \(L^\infty\) spaces for \(0\le\beta\le1/4\) and ill-posedness in these spaces for \(1/4<\beta\le1\). They also identified \(\beta=1/4\) as the boundary between finite and infinite capacity for the inverse wave-action cascade [AL25, Remark 1.5]. For power-law dispersion \(\omega(k)=|k|^a\), \(1\le a\le5\), Zhu [Zhu26] obtained sharp local well-posedness and ill-posedness criteria in weighted \(L^\infty\) spaces, depending on the interaction exponent and the decay of the initial data.

Condensation has also been studied for the closely related Boltzmann–Nordheim equation for Bose gases. A foundational theory of global isotropic measure solutions and their condensation behavior was developed by Lu [Lu04, Lu13]. A central contribution was made by Escobedo and Velázquez [EV14, EV15a], who proved finite-time blow-up for classical solutions and finite-time formation of a condensate for weak solutions, including the supercritical regime in which the corresponding Bose–Einstein equilibrium contains a Dirac mass. Cai and Lu [CL19, CL26] subsequently studied convergence to Bose–Einstein equilibrium and the influence of the interaction potential on condensation. More recently, under scattering assumptions with \(0\le\eta<1\), Luo and Liu [LL26] proved that every admissible initial measure in the full subcritical regime \(\overline T/\overline T_c<1\) admits a conservative isotropic solution that develops a condensate in finite time, without any initial low-energy concentration assumption.

In addition to 4-wave equations, 3-wave kinetic models have also been studied in several settings, including phonon interactions, stratified ocean flows, capillary and acoustic waves, and Bose gases [Spo06, GST20, NT18, TCSB20, ST18, ST20]. Related existence, well-posedness, condensation, and energy-transfer phenomena have been investigated in these works. More recently, Staffilani and Tran [ST25] studied a kinetic model containing both 3-wave and 4-wave interaction terms and proved finite-time and immediate energy cascades for suitable classes of initial data.

The rigorous derivation of wave kinetic equations from nonlinear dispersive equations has also been studied as part of Hilbert’s sixth problem. Early results concerned equilibrium correlations for the lattice nonlinear Schrödinger equation [LS11] and kinetic approximations for the cubic nonlinear Schrödinger equation on time scales shorter than the kinetic time [BGHS21, DH21, CG25, CG26]. Deng and Hani [DH23a] derived the homogeneous 4-wave kinetic equation from the cubic nonlinear Schrödinger equation with random initial data in dimensions \(d\ge3\), up to a sufficiently small multiple of the kinetic time. They considered the joint large-box and weak-nonlinearity limit under a particular scaling law. Their subsequent work covered the full range of admissible scaling laws [DH26] and extended the approximation to every compact interval of rescaled time within the lifespan of the smooth kinetic solution [DH23b]. Staffilani and Tran [ST21] justified a 3-wave kinetic limit, in the resonance broadening sense, for a stochastic Zakharov–Kuznetsov equation on a lattice. Related work treats spatially inhomogeneous models [ACG25, HRST22] and damped–driven nonlinear Schrödinger equations [DK21, GH26].

Organization of the paper

The rest of the paper is organized as follows. In Section 2, we develop the framework for measure solutions and prove Theorem 1.5. Section 3 proves that the condensate mass remains constant for \(1/4\le\beta\le3/4\). Section 4 establishes density inequalities and a comparison principle for \(0<\beta<1/4\). Using these estimates, Section 5 proves finite-time condensation and convergence to \(\mathcal N_0\delta_0\) for initial measures with a nonzero absolutely continuous part when \(0<\beta<1/4\), completing the proof of Theorem 1.6.

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