Regularity of the Vlasov–Poisson–Boltzmann System Without Angular Cutoff

Smoothing Estimates of the Vlasov-Poisson-Landau System

“…This result shows that the smoothing effect proven (for example in [2]) for the Landau equation extends to Vlasov-Poisson-Landau system for all variables t, x, v. This behavior was also observed for Boltzmann equation [1,3] and Vlasov-Poisson-Boltzmann system [5]. Notice that the smoothing property holds uniformly in Theorem 1.2 when the time goes to infinity.…”

“…Since we are dealing with Vlasov-Poisson-Landau system, the idea here is similar to the Boltzmann equation case [8]. Similar macroscopic estimate can be found in [5,17]. Notice that the calculation in this section is valid for both hard potential γ + 2 ≥ 0 and soft potential γ + 2 < 0.…”

“…) dv * , with Φ defined by (5). For any vector-valued function h(v) = (h 1 (v), h 2 (v), h 3 (v)), we define projection…”

Global Regularity of the Vlasov-Poisson-Boltzmann System Near Maxwellian Without Angular Cutoff for Soft Potential

“…We refer to [9,16] for the Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equation. Recently, the author [14,15] established the smoothing effect of the Cauchy problem for VPB system with hard potential and VPL system for Coulomb interactions. These works show that the Boltzmann operator behaves locally like a fractional operator Q( f ,g) ∼ (−∆ v ) s g+lower order terms.…”

“…Since Duan and Liu [17] found the global solution for non-cutoff soft potential with 1/2 ≤ s < 1, the smoothing effect for the VPB system is an open interesting problem. In [14], the author finds out the smoothing effect for hard potential. In this work, we finally recover the smoothing effect for non-cutoff soft potential with the whole range 0 < s < 1.…”