Stability of vacuum for the Boltzmann Equation with moderately soft potentials

Abstract

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials and any singularity parameter $s\in (0,1)$, i.e. with $\gamma +2s\in (0,2]$ on the whole space ${\mathbb{R}}^3$. We prove that if the initial data $f_{\text{in}}$ are close to the vacuum solution $f_{\text{vac}}=0$ in an appropriate weighted norm then the solution $f$ remains regular globally in time and approaches a solution to a linear transport equation. Our proof uses $L^2$ estimates and we prove a multitude of new estimates involving the Boltzmann kernel without angular cut-off. Moreover, we rely on various previous works including those of Gressman--Strain, Henderson--Snelson--Tarfulea and Silverstre. From the point of view of the long time behavior we treat the Boltzmann collisional operator perturbatively. Thus an important challenge of this problem is to exploit the dispersive properties of the transport operator to prove integrable time decay of the collisional operator. This requires the most care and to successfully overcome this difficulty we draw inspiration from Luk's work [Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11] and that of Smulevici [Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE, 2(2):Art. 11, 55, 2016]. In particular, to get at least integrable time decay we need to consolidate the decay coming from the space-time weights and the decay coming from commuting vector fields.

arXiv comment82 pages. Updated to published version. Corrected many typos and a different simplified proof for Lemma 6.6

References in corpus (14)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Global Strong Solutions of the Boltzmann Equation without Angular Cut-off
• Regularizing effect and local existence for non-cutoff Boltzmann equation
• Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production
• A vector field method for relativistic transport equations with applications
• Decay estimates for large velocities in the Boltzmann equation without cutoff
• Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section
• Sharp asymptotic behavior of solutions of the $3d$ Vlasov-Maxwell system with small data
• A commuting-vector-field approach to some dispersive estimates
• Global Solutions of the Boltzmann Equation over ${\mathbb{R}}^D$ near Global Maxwellians with Small Mass
• A vector field method for massless relativistic transport equations and applications
• Stability of vacuum for the Boltzmann Equation with moderately soft potentials
• Well-posedness and scattering for the Boltzmann equations: Soft potential with cut-off
• Asymptotic properties of the solutions to the Vlasov-Maxwell system in the exterior of a light cone

Cited by in corpus (1)

• Stability of vacuum for the Boltzmann Equation with moderately soft potentials