Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation

Abstract

We present in this document the Lebesgue and Sobolev propagation of exponential tails for solutions of the homogeneous Boltzmann equation for hard and Maxwell interactions. In addition, we show the $L^p$-integrability creation of such tails in the case of hard interactions. The document also presents a result on exponentially-fast convergence to thermodynamical equilibrium and propagation of singularities and regularization of such solutions. All these results are valid under the mere Grad's cut-off condition for the angular scattering kernel. Highlights of this contribution include: (1) full range of $L^p$-norms with $p\in [1,\infty ]$, (2) analysis for the critical case of Maxwell interactions, (3) propagation of fractional Sobolev exponential tails using pointwise conmutators, and (4) time asymptotic and propagation of regularity and singularities under general physical data. In many ways, this work is an improvement and an extension of several classical works in the area; we use known techniques and introduce new and flexible ideas that achieve the proofs in an elementary manner.

Improved overall presentation, added comments for context, updated references, addition of Lemma 4.1 and Remark 5.1, sharper constants in Section 5 for better spectral gap estimation

References in corpus (6)

• Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials
• Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions
• Explicit coercivity estimates for the linearized Boltzmann and Landau operators
• Convolution inequalities for the Boltzmann collision operator
• On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
• Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations

Cited by in corpus (3)

• Decay estimates for large velocities in the Boltzmann equation without cutoff
• Burnett Spectral Method for High-Speed Rarefied Gas Flows
• Stability, well-posedness and regularity of the homogeneous Landau equation for hard potentials