Linearized Boltzmann Equation Analysis

• Linearized Boltzmann Equation is a fundamental kinetic theory tool that characterizes small perturbations around thermodynamic equilibrium using spectral decomposition on periodic domains.
• The analysis separates fluid-like modes, exhibiting algebraic decay linked to hydrodynamic behavior, from kinetic modes that decay exponentially due to collision effects.
• Iterative Picard methods combined with the mixture lemma facilitate the transfer of regularity, enabling sharp space-time estimates and improved handling of non-smooth initial data.

The linearized Boltzmann equation is a fundamental tool in kinetic theory, providing a rigorous framework to analyze perturbations around thermodynamic equilibrium for dilute gases, including the intricate interplay between fluid-like and kinetic behaviors. Its paper on periodic domains, such as the 3-dimensional torus, reveals both deep spectral structures and precise space-time dynamics, particularly relevant when initial data are non-smooth. Notably, this analysis uncovers the coexistence of slowly decaying fluid waves and rapidly attenuating kinetic modes, with decay rates modulated by the geometry and size of the domain.

1. Fluid Regime and Spectral Decomposition

The linearized Boltzmann equation on a torus exhibits a clear separation between fluid-like and kinetic-like components, reflecting the kernel structure of the collision operator $L$. The fluid-like part corresponds to the projection of the solution onto the span of the collision invariants, i.e., the five-dimensional space generated by density, momentum, and energy, and is associated with long-wavelength (low-frequency) Fourier modes. For each spatial frequency $k$, the Fourier-transformed operator $-i\pi\varepsilon\,\xi\cdot k + L$ yields a spectrum in which, for $|\varepsilon k|$ small, five eigenvalues bifurcate from the origin. These encode collective hydrodynamic behavior—manifest in wave propagation and dissipative fluid dynamics analogous to the compressible Navier–Stokes system with size-dependent dissipation.

Explicitly, the solution can be organized by the Green function: $$G_t(x, \xi) = \sum_{k\in\mathbb{Z}^3} e^{i\pi\varepsilon k\cdot x + \left( -i\pi\varepsilon\,\xi\cdot k + L \right)t }$$ Decomposing into long-wave ($|\varepsilon k| < \delta$) and short-wave ($|\varepsilon k| \geq \delta$) contributions isolates the fluid regime and its spectral features.

2. Kinetic Regime and Picard Iteration

Complementing the fluid component, the kinetic-like part comprises all modes orthogonal to the five collision invariants. These modes are governed primarily by the transport operator and retain potential spatial non-smoothness in initial data. The kinetic evolution is constructed via a Picard-type iterative expansion: $f = h^{(-1)} + h^{(0)} + h^{(1)} + \cdots$ where $h^{(-1)}$ solves the damped transport equation: $$\partial_t h^{(-1)} + \xi\cdot\nabla_x h^{(-1)} + v(\xi) h^{(-1)} - K_s h^{(-1)} = 0$$ and for $j\geq 0$,

$$\partial_t h^{(j)} + \xi\cdot\nabla_x h^{(j)} + v(\xi) h^{(j)} = K h^{(j-1)}, \quad h^{(j)}(x,0,\xi)=0$$

with $K_s$ and $K_r$ the singular and regular components of the collision smoothing operator. This approach systematically increases the space regularity of $f$ due to the regularizing properties of $K$, and, crucially, provides for exponential time decay of the kinetic contribution, governed by the spectral gap in $v(\xi)$.

3. Domain-Size-Dependent Time Decay Rates

The time decay of each solution component is quantitatively distinct. The fluid-like modes decay with rates dependent on the torus diameter (through $\varepsilon$): for small $\varepsilon$ (approaching the whole-space limit), the $L^2$ decay of the fluid part is $O((1+t)^{-3/2})$. The spectral analysis shows that this slow decay arises from the proximity to the continuous spectrum associated with translation invariance—a behavior absent in bounded or highly confined domains, where decay becomes more rapid. In contrast, all short-wave (kinetic) modes decay at a uniform exponential rate, $O(e^{-\nu_0 t})$, due to the dissipativity of the collision term.

This dichotomy underlies the critical domain-size effect on long-time relaxation: large domains exhibit persistent, slowly decaying hydrodynamic waves, whereas smaller ones promote faster equilibration.

4. The Mixture Lemma and Regularity Transfer

A pivotal result in the propagation of regularity is the mixture lemma, which enables the transfer of velocity smoothness to space and time by exploiting the commutation properties of differential operators: $$\mathcal{D}_t = t\,\nabla_x + \nabla_\xi, \qquad [\mathcal{D}_t,\;\partial_t + \xi\cdot\nabla_x] = 0$$ This operator commutes with the free transport generator, so that derivatives in $\xi$ can be replaced with mixed derivatives in $x$ and $t$. Through this device, and application of a carefully constructed energy estimate,

$$\| M_j f_0 \|_{L^2_x H^2_\xi} \leq Ce^{-(\nu_0/3)t} \|f_0\|_{L^2_x H^2_\xi},\qquad (j=1,2)$$

the regularizing effect of the collision kernel is systematically transferred to macroscopic variables in the kinetic evolution, without constructing explicit solutions to the (damped) transport equation. This framework is broadly adaptable, including to Landau and Fokker–Planck-type operators.

5. Mathematical Structure and Formulas

The key mathematical formulations driving the solution structure include:

• Fourier and Green function representations separating fluid and kinetic scales:

$$G_t^{L}(x,\xi) = \sum_{|\varepsilon k|<\delta} e^{i\pi\varepsilon k\cdot x + ( -i\pi\varepsilon\,\xi\cdot k + L)t }$$

• Fluid mode expansion in terms of eigenvalues and eigenfunctions $e_j(\varepsilon k)$ of $-i\pi\varepsilon\xi\cdot k+L$ and initial data projections $\hat{I}_k$.
• Picard iterative scheme for the kinetic part, leveraging the regularizing action of $K$ by iteration in time.
• Use of commuting differential operators for the transfer of $\xi$-regularity to $(x, t)$-regularity.

These structures underpin the extraction of both long-wave hydrodynamics and exponentially decaying kinetic modes.

6. Broader Significance and Extensions

The described methodology not only achieves a rigorous description of the pointwise behavior of the linearized Boltzmann equation on the torus but also establishes robust analytical tools for handling models that couple fluid and kinetic effects. The spectral long-wave analysis exposes the precise connection between kinetic theory and macroscopic fluid equations, providing a direct explanation for domain-dependent relaxation rates. The iterative (Picard) construction, combined with the mixture lemma, yields a flexible mechanism to handle non-smooth initial data while guaranteeing improved regularity and quantitative decay.

These advances facilitate extensions to a range of kinetic equations (e.g., linearized Landau, Fokker–Planck), inform the development of sharp space-time estimates in nonlinear settings, and clarify the fluid-to-kinetic transition in bounded or periodic geometries. The analytical framework established forms a rigorous foundation for subsequent studies in both mathematical kinetic theory and computational applications, where understanding the interplay between transport, collision, and domain geometry is critical for modeling relaxation, wave propagation, and the emergence of hydrodynamics in rarefied gases.