[Paper Review] Spectral and phase space analysis of the linearized non-cutoff Kac collision operator

Nicolás Lerner, Yoshinori Morimoto+2 more|arXiv (Cornell University)|Nov 2, 2011

Gas Dynamics and Kinetic Theory19 references3 citations

TL;DR

This paper provides a spectral and phase space analysis of the linearized non-cutoff Kac collision operator, showing it is a fractional power of the harmonic oscillator and diagonal in the Hermite basis. It establishes a complete asymptotic expansion for its symbol in a symbolic calculus class, extending results to the linearized non-cutoff radially symmetric Boltzmann operator.

ABSTRACT

The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven.

Motivation & Objective

• To analyze the spectral and microlocal properties of the linearized non-cutoff Kac collision operator in the context of kinetic theory.
• To establish that the linearized Kac operator is a function of the harmonic oscillator and diagonal in the Hermite basis.
• To derive a complete asymptotic expansion for the symbol of the linearized operator in a class with a well-behaved symbolic calculus.
• To extend the results to the linearized non-cutoff radially symmetric Boltzmann operator via analogous analysis.

Proposed method

• Use of the Weyl quantization to relate the operator to its symbol in phase space, enabling microlocal analysis.
• Application of the Hermite basis expansion to diagonalize the linearized Kac operator, revealing its structure as a function of the harmonic oscillator.
• Derivation of asymptotic expansions for the symbol of the operator using integral estimates involving angular singularities and Bessel-type integrals.
• Employment of radial symmetry and Borel’s theorem to reduce the analysis to radial functions and their Fourier transforms.
• Use of weak formulations of the Weyl quantization via Wigner functions to define operators for tempered distributions.
• Analysis of the even part of distribution kernels to relate operators acting on even functions to their symmetrized symbols.

Experimental results

Research questions

• RQ1 How does the linearized non-cutoff Kac operator relate to the harmonic oscillator in spectral terms?
• RQ2 What is the complete asymptotic expansion of the symbol of the linearized Kac operator in a symbolic calculus class?
• RQ3 Can the spectral structure of the linearized Kac operator be extended to the linearized non-cutoff radially symmetric Boltzmann operator?
• RQ4 What role does the non-integrable angular singularity play in the smoothing properties of the operator?
• RQ5 How does the Weyl quantization framework facilitate the phase space analysis of the operator?

Key findings

• The linearized non-cutoff Kac operator is shown to be a function of the harmonic oscillator and diagonal in the Hermite basis.
• The operator is identified as a fractional power of the harmonic oscillator, with the exponent depending on the singularity strength $ s $ in the angular factor.
• A complete asymptotic expansion is established for the symbol of the linearized Kac operator in a class with a well-defined symbolic calculus.
• The eigenvalues $ ilde{ u}_l $ of the operator decay exponentially as $ ilde{ u}_l o 0 $ when $ l o lat $, with the bound $ ilde{ u}_l o 0 $ like $ rac{4^{2s} au}{1-s} au^{-2l au} $ for $ l o lat $, where $ au = au rac{4}{ au} $.
• The symbol of the linearized Kac operator admits a full asymptotic expansion in terms of homogeneous symbols, reflecting the operator's pseudodifferential nature.
• The results are extended to the linearized non-cutoff radially symmetric Boltzmann operator, establishing analogous spectral and symbolic properties.