Measure valued solution to the spatially homogeneous Boltzmann equation with inelastic long-range interactions

Abstract

This paper aims at studying the inelastic Boltzmann equation without Grad’s angular cutoff assumption, where the well-posedness theory of the Cauchy problem is established for the Maxwellian molecules in a space of probability measure defined by Cannone and Karch [Commun. Pure Appl. Math. 63, 747–778 (2010)] via Fourier transform and the infinite energy solutions are not a priori excluded. The key strategy is to construct a brand new geometric relation of the inelastic collision mechanism to extend the result of Cannone and Karch from moderate singularity of the non-cutoff collision kernels to strong singularity and simultaneously handle more general restitution coefficients. Moreover, we extend the self-similar solution to the Boltzmann equation with infinite energy shown by Bobylev and Cercignani [J. Stat. Phys. 106, 1039–1071 (2002)] to the inelastic case by using a constructive approach, which is also proved to be the large-time asymptotic steady solution with the help of an asymptotic stability result in a certain sense.