Botzmann Equation

Smoothing Properties of the Boltzmann Equation

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian , which has led to the conjecture that the weak solution of the non-cutoff homogenous Boltzmann equation with initial datum having finite mass, energy and entropy, should immediately become Gevrey regular. This is in sharp contrast to the so called Grad angular cut-off case, where the singularity in the angular collision kernel \(b(\cos\theta)\) for small angles, that is, grazing collisions, is artificially removed.

The question of regularity of solutions is closely related to the convergence to equilibrium as time \(t\to\infty\), as highlighted in a theorem of Desvillettes and Villani , which shows convergence to equilibrium for the Boltzmann equation, is conditional in the sense that it needs as an input that solutions of the Boltzmann equation are highly regular (measured in the scale of Sobolev spaces). So there is a need to study the regularity, in particular the smoothing, of solutions of the Boltzmann equation.

We study optimal smoothing properties and were able to prove the Gevrey smoothing conjecture of Desvilettes and Wennberg in the homogeneous case for Maxwellian molecules (see [2]), and a similar smoothing property for the homogeneous Boltzmann equation with a Debye-Yukawa type interaction in (see [1]). Our proof relies on a quasi-locality property in Fourier space of the collision operator, which enables the construction of an inductive scheme controlling the non-local and non-linear commutation error of the Boltzmann operator with an exponentially (more precisely, sub-Gaussian) growing weight in Fourier space.

Related publications

  1. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic and Related Models 10 (2017), 901-924.
    (with J.-M. Barbaroux, D. Hundertmark, S. Vugalter)
  2. Gevrey smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules. Archive for Rational Mechanics and Analysis 225 (2017), 601–661.
    (with J.-M. Barbaroux, D. Hundertmark, S. Vugalter)