Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff

The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

“…In this part, we first establish an essential coercivity estimate of the linearized collision operator, in the framework of general cross sections. As shown in [7,8] for the special Maxwellian case, this estimate will play an important role for the related Cauchy problem.…”

The Boltzmann Equation Without Angular Cutoff in the Whole Space: Ii, Global Existence for Hard Potential

“…This paper is among the series works on the Boltzmann equation with non-angular cutoff cross-section and it follows the paper [7] (herein referred as Part I), extending our initial work [5,6] on the same problem for Maxwellian molecule. Consider (1.1)…”

“…In order to avoid the unnecessary repetition, readers can refer to Part I, comments and references. Here, we just refer the references [9,10,11,17,18,19] for the general background of the Boltzmann equation and the recent progress on the mathematical theories for the case without angular cutoff, [1,2,3,4,5,6,7,8,14,15,16]. Hence, we now directly go to the Cauchy problem for the perturbation denoted by g = µ − 1 2 ( f − µ)…”

Bounded solutions of the Boltzmann equation in the whole space

“…The same space was used also in [14]. On the other hand, it is clear that the space H k,ℓ ul (R 6 ) defined by (1.4), the locally uniform Sobolev space with respect to x-variables, contains, as its subset, not only the spaces H k ℓ (R 6 ) and H k ℓ (T 3 × R 3 ) but also the set of functions having the form G = µ + µ 1/2 g. If µ is a global Maxwellian, then we have well-known perturbative solutions of equilibrium as in [5,6,7,22], but more generally G can be any bounded functions. Hence, Theorem 2.1 gives, for example, almost periodic solutions, solutions having different limits at x-infinity like shock profile solutions which attain different equilibrium at the right and left infinity, and bounded solution behaving in more general way at x-infinity.…”

“…An essential observation here is that so far, all solutions for the Cauchy problem have been constructed so as to satisfy one of the following three spatial behaviors at infinity; x-periodic solutions (solutions on the torus, [13,14,21]), solutions approaching an equilibrium ( [4,5,6,7,15,19,22]) and solutions approaching 0 (solutions near vacuum, [1,3,9,12]). Notice that the solutions constructed in [16] are also solutions approaching an global equilibrium.…”