Local well-posedness of the Boltzmann equation with polynomially decaying initial data

Abstract

We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed L 2 and L ∞ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.

Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

“…Finally, we have a simple interpolation lemma that allows us to trade decay for regularity. The proof is the same as [12,Lemma 2.6].…”

“…We need a short-time existence theorem that allows initial data to decay only polynomially in v (rather than exponential or Gaussian decay). This was established in [21] for s ∈ (0, 1 2 ) and γ ∈ (− 3 2 , 0], and in [12] for s ∈ (0, 1) and γ ∈ (max{−3, − 3 2 − 2s}, 0), but these results do not apply to the case γ > 0. Here, we provide a relatively quick proof of short-time existence when the initial data is near a Maxwellian, that applies both for γ ≤ 0 and γ > 0.…”

“…It requires the initial data to have Gaussian decay for large velocities, which makes it difficult to apply in practice. In [21] and [12], the authors obtain a short-time existence result for initial data in H 6 and H 5 respectively, with a polynomial weight, in the case of soft potentials: γ ≤ 0.…”

“…Once we have the three ingredients mentioned above, proving Theorems 1.1 and 1.2 becomes practically trivial. Using the result of [12], we could have shortened the document considerably, but we wanted to be able to address the case of hard potentials, γ > 0, as well. The proof for short-time existence we include here is relatively minimalistic, and it is not meant to be applied to initial data that is far from equilibrium.…”

“…Such a conditional regularity result could be used to extend the result of Theorems 1.1 and 1.2 to the same values of γ and s. The short-time existence result we present in Section 3 also uses the restriction γ + 2s ≥ 0 indirectly through the application of the result in [11] for the construction of solutions. Note that, for 0 > γ > max(−3, −3/2 − 2s), we can use the result in [12] to construct the solutions for a short time.…”