A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation

Abstract

The purpose of this paper is to introduce a new characterization of the characteristic functions for the study on the measure valued solution to the homogeneous Boltzmann equation so that it precisely captures the moment constraint in physics. This significantly improves the previous result by Cannone and Karch (2010) [2] in the sense that the new characterization gives a complete description of infinite energy solutions for the Maxwellian cross section. In addition, the global in time smoothing effect of the infinite energy solution is justified as for the finite energy solution except for a single Dirac mass initial datum.

Convergence to self-similar solutions for the homogeneous Boltzmann equation

“…The relation between P α (R 3 ) and K α (R 3 ) was given in [7] and [12] and it can be stated as follows.…”

“…From now on, we consider the Cauchy problem for (1.6) with initial condition ϕ(0, ξ) = ϕ 0 (ξ). (1.9) For α ∈ (2s, 2], it is shown in [7,11,12] that this Cauchy problem admits a unique global solution ϕ(t, ξ) ∈…”

Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

“…It was proved in [20] that the smoothing effect of measure valued solutions always occurs except for a single Dirac mass initial datum. After that, the results about the existence and smoothing effect of solutions have been improved in [18,19] by introducing more precise characteristic function spaces M α and M α , which satisfy relations…”

The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand–Shilov smoothing effect

“…Recently, Morimoto, Wang and Yang in [14] introduce a new classification on the characteristic functions and prove the smoothing effect under this measure initial datum, see also [15] and [2]. For the Shubin space, we have…”