Gevrey Smoothing for Weak Solutions of the Fully Nonlinear Homogeneous Boltzmann and Kac Equations Without Cutoff for Maxwellian Molecules

Abstract

Abstract. It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in Le., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

“…Remark 1.5. This regularity is much weaker than the Gevrey regularity we proved in [4] for singular kernels of the form (3), but it is much stronger than the H ∞ smoothing shown in [11]. Moreover, it is exactly the right type of regularity one would expect for a coercive term of the form (7) from the analogy with the heat equation (11).…”

“…starting with arbitrary initial datum f 0 ≥ 0, f 0 ∈ L 1 2 ∩ L log L, one has f (t, ·) ∈ H ∞ for any positive time t > 0. Based upon our recent proof [4] of Gevrey smoothing for the homogeneous Boltzmann equation with Maxwellian molecules and angular singularity of the inverse-power law type (3), we can show a stronger than H ∞ regularisation property of weak solutions in the Debye-Yukawa case.…”

“…In this case, the coercive effects are much weaker and of the form −Q(g, f ) ≈ (log(1 − ∆)) µ+1 f + lower order terms, (7) as was noticed in [11], see also appendix A. In this work, as in [4,11], we consider only the socalled Maxwellian molecules approximation, where the collision kernel does not depend on v − v * , but only on the collision angle θ.…”

“…(1) For the weights needed in the proof of Theorem 1.4 we have a much stronger enhanced subadditivity bound, see Lemma 2.1. The proof is more involved than the one in [4], though. (2) Because of the stronger form of the subadditivity bound, we can allow for a bigger loss in the induction step.…”

“…For non-power-type Fourier multipliers this commutator is considerably more complicated than the one encountered in the H ∞ smoothing case. To overcome this, we follow the strategy we developed in [4], where an inductive procedure was invented to control the commutation error, in order to prove the Gevrey smoothing conjecture in the Maxwellian molecules case.…”