Bounded solutions of the Boltzmann equation in the whole space

Abstract

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.2000 Mathematics Subject Classification. 35A01, 35A02, 35A09, 35S05, 76P05, 82C40.

Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

“…Next, we extend our continuation criterion from [21] to the solutions of Theorem 1.2, and sharpen it in the case of very soft potentials: Theorem 1. 3. The solution f constructed in Theorem 1.2 can be extended for as long as the quantity Ψ remains finite, where p > 3/(3 + γ) and p = ∞ for γ = −3.…”

The Boltzmann Equation Without Angular Cutoff in the Whole Space: Qualitative Properties of Solutions

“…In [11], bounded solutions of the Boltzmann equation in the whole space have been constructed without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, it has been shown that if the initial data is non-negative and belongs to a uniformly local Sobolev space with the Maxwellian decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity.…”

“…Let us recall that in a close to equilibrium framework, the existence of such classical solutions was proven in our series of papers [9,10] and using a different method, by Gressmann and Strain [22,23,24]. We refer also to [11] for bounded local solutions.…”

Local existence, lower mass bounds, and a new continuation criterion for the Landau equation

“…Note that, in light of the definition of H k,l ul , Theorem 1.1 makes no assumption on the behavior of f in as |x| → ∞. The requirement that f in have four Sobolev derivatives is an improvement over [27], and matches the current state of the art for results on the Boltzmann equation [3,4,5]. At this time, it is unclear whether this hypothesis can be relaxed further.…”

“…Such an approximation is not available for the Landau equation because the Landau equation results from focusing on grazing collisions in the Boltzmann equation, i.e. taking the limit where the angular singularity essentially becomes a derivative in v. We point out that our proof covers all cases γ ∈ [−3, 0), which requires extra care, while [4,5] make the restriction that γ > −3/2, and [3] replaces the factor |v − w| γ in the Boltzmann collision kernel with (1 + |v − w| 2 ) γ/2 , which also sidesteps the difficulties associated with very soft potentials.…”