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

Abstract

International audienceWe study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation

Shubin Regularity for the Radially Symmetric Spatially Homogeneous Boltzmann Equation with Debye-Yukawa Potential

“…one has f (t, ·) ∈ H ∞ (R 3 ) for any positive time t > 0. This result was extended by J.-M. Barbaroux, D. Hundertmark, T. Ried, S. Vugalter in [2]. They showed a stronger regularisation property : for any 0 < s < 2, and for any T 0 > 0, there exist β, M > 0 such that…”

Regularity of Non-cutoff Boltzmann Equation with Hard Potential

“…Later on, many works discover the optimal regular estimate of Boltzmann collision operator in v in different setting. We refer to [2,9,11,24,31] for the dissipation estimate of collision operator, and [3,7,8,10,12,13,[18][19][20][25][26][27] for smoothing effect of the solution to Boltzmann equation in different aspect. These works show that the Boltzmann operator behaves locally like a fractional operator:…”

Exponential Convergence to Equilibrium for Solutions of the Homogeneous Boltzmann Equation for Maxwellian Molecules

“…In fact, it is also used as a toy model to understand some mathematical properties of more complex equations (such as the inhomogeneous Boltzmann equation), for which the same properties prove to be unsolved questions. See, e.g., [6][7][8][9][10].…”