论文标题
真空对具有中等柔软电势的玻尔兹曼方程的稳定性
Stability of vacuum for the Boltzmann Equation with moderately soft potentials
论文作者
论文摘要
我们在空间上考虑具有适度柔软潜力的空间非均匀性非固定型玻尔兹曼方程（0,1）$中的任何奇异性参数$ s \ in（0,1）$，即$γ+2s \ in（0,2] in（0,2] $（in（0,2] $真空解决方案$ f _ {\ text {vac}} = 0 $在适当的加权标准中，解决方案$ f $在全球范围内保持了定期，并使用了线性传输方程的解决方案。亨德森（Henderson） - 纳尔逊（Snelson） - 从长期行为的角度来看，我们对鲍尔茨曼（Boltzmann）碰撞运算师的看法是，这种问题的重要挑战是利用运输运营商的范围来证明这一艰难的态度。 PDE（2019）5:11的Landau方程和smulevici的年鉴[vlasov-poisson系统的小数据解决方案和矢量场方法，PDE，2（2）。向量字段。
We consider the spatially inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials and any singularity parameter $s\in (0,1)$, i.e. with $γ+2s\in(0,2]$ 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.
