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Smoothing Properties of a Linearization of the Three Waves Collision Operator in the bosonic Boltzmann-Nordheim Equation

Jogia Bandyopadhyay, Jani Lukkarinen

TL;DR

The paper analyzes the linearized dynamics of the bosonic Boltzmann–Nordheim equation around a condensate-containing equilibrium, focusing on the dominant three-wave collision operator $L_3$ which carries a point singularity at the origin and a line singularity along $x=y$. It develops a dual framework: (i) a weighted $L^2$ theory establishing a Friedrichs extension $\overline L_3$, non-negativity, a spectral gap, and contractive semigroups; and (ii) a Banach-space approach with a time-dependent Hölder-type weight $\Gamma_t$ to prove smoothing of the two-variable difference $\Delta_t(x,y) = \psi_t(x)-\psi_t(y)$. The analysis uses regularized operators $L_3^{\varepsilon}$ and a detailed Duhamel formulation to connect $\Delta_t$ with regularized difference solutions $D\psi_t^{\varepsilon}$, obtaining uniform-in-$\varepsilon$ estimates and a convergence result, which yields Hölder regularity improvement for finite time. The results provide a rigorous smoothing mechanism for the linearized evolution, supporting nonlinear perturbation arguments for the Boltzmann–Nordheim dynamics with condensates and informing the long-time behavior toward equilibrium. Altogether, the work establishes contractive semigroups, spectral gaps, and finite-time Hölder smoothing for the linearized operator, forming a solid foundation for nonlinear stability analyses in this quantum kinetic setting.

Abstract

We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non-equilibrium state which includes both normal fluid and a condensate. More precisely, we look at the previously postulated nonlinear Boltzmann-Nordheim equations for such systems, in a spatially homogeneous state which has an isotropic momentum distribution, and we linearize the equation around an equilibrium state which has a condensate. We study the most singular part of the linearized operator coming from the three waves collision operator for supercritical initial data. The operator has two types of singularities, one of which is similar to the marginally smoothing operator defined by the symbol $\ln(1+p^2)$. Our main result in this context is that for initial data in a certain Banach space of functions satisfying a H\"{older} type condition, at least for some finite time, evolution determined by the linearized operator improves the Hölder regularity. The main difficulty in this problem arises from the combination of a point singularity and a line singularity present in the linear operator, and we have to use certain fine-tuned function spaces in order to carry out our analysis.

Smoothing Properties of a Linearization of the Three Waves Collision Operator in the bosonic Boltzmann-Nordheim Equation

TL;DR

The paper analyzes the linearized dynamics of the bosonic Boltzmann–Nordheim equation around a condensate-containing equilibrium, focusing on the dominant three-wave collision operator which carries a point singularity at the origin and a line singularity along . It develops a dual framework: (i) a weighted theory establishing a Friedrichs extension , non-negativity, a spectral gap, and contractive semigroups; and (ii) a Banach-space approach with a time-dependent Hölder-type weight to prove smoothing of the two-variable difference . The analysis uses regularized operators and a detailed Duhamel formulation to connect with regularized difference solutions , obtaining uniform-in- estimates and a convergence result, which yields Hölder regularity improvement for finite time. The results provide a rigorous smoothing mechanism for the linearized evolution, supporting nonlinear perturbation arguments for the Boltzmann–Nordheim dynamics with condensates and informing the long-time behavior toward equilibrium. Altogether, the work establishes contractive semigroups, spectral gaps, and finite-time Hölder smoothing for the linearized operator, forming a solid foundation for nonlinear stability analyses in this quantum kinetic setting.

Abstract

We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non-equilibrium state which includes both normal fluid and a condensate. More precisely, we look at the previously postulated nonlinear Boltzmann-Nordheim equations for such systems, in a spatially homogeneous state which has an isotropic momentum distribution, and we linearize the equation around an equilibrium state which has a condensate. We study the most singular part of the linearized operator coming from the three waves collision operator for supercritical initial data. The operator has two types of singularities, one of which is similar to the marginally smoothing operator defined by the symbol . Our main result in this context is that for initial data in a certain Banach space of functions satisfying a H\"{older} type condition, at least for some finite time, evolution determined by the linearized operator improves the Hölder regularity. The main difficulty in this problem arises from the combination of a point singularity and a line singularity present in the linear operator, and we have to use certain fine-tuned function spaces in order to carry out our analysis.
Paper Structure (31 sections, 41 theorems, 400 equations)

This paper contains 31 sections, 41 theorems, 400 equations.

Table of Contents

  1. Introduction
  2. Physical motivation of the nonlinear problem and the proposed linearization
  3. Solutions in a Weighted $L^2$-space
  4. Friedrichs Extensions and Their Properties
  5. Approximating the Semigroup Solution generated in $L^2(\nu)$ by $\overline L_3$
  6. Smoothing Solutions in a weighted Banach Space of Hölder-continuous functions
  7. Existence and Uniqueness of Solutions in a Banach space with a Time-dependent Hölder-type Condition
  8. Derivation of Equation (\ref{['eq: DeltaEvol']})
  9. Solutions of (\ref{['eq: DeltaEvol']}): The proof of Theorem \ref{['theorem: DeltaSolutionsY']}
  10. Solutions of the Regularized Evolution Equation:
  11. Regularized Solution $\psi^{\varepsilon}_t$
  12. Regularized Evolution Equation for the Difference Variable $D\psi^{\varepsilon}_t$ and the Existence of a Unique Solution
  13. Connecting the Variables $\Delta_t$ and $D\psi^{\varepsilon}_t$: Evolution Equation for $D_t=D\psi^{\varepsilon}_t-\Delta_t$
  14. Linearization of the Three-wave Collision Operator $\mathcal{C}_3$
  15. Parameters $\alpha, c_0$ and $\mu$ appearing in the Weight Functions for the Banach Spaces
  16. ...and 16 more sections

Key Result

Lemma 2.1

$D(Q)$ is complete under the inner product $Q$.

Theorems & Definitions (76)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 66 more