Optimal time-decay rates of the Boltzmann equation

Spectrum Structure and Behaviors of the Vlasov--Maxwell--Boltzmann Systems

“…For twospecies VMB system, we can observe some phenomena of wave propagation and magnetic field domination on long time behaviors of charge transport of charge transport due to the effect of magnetic filed and mutual interaction between the particles of two species. Indeed, for the global solution (f 1 , f 2 , E, B) to the linearized two-species VMB system, we can show that the distribution function f 1 , corresponding to the total summation of the distribution functions between the two species, is governed by the linearized Boltzmann equation and its optimal time decay rate (1 + t) − 3 4 in L 2 -norm has been already established for instance in [20,25]. Meanwhile, the magnetic field B is also shown to tend to zero at the optimal time decay rates (1 + t) − 3 4 in L 2 -norm, but the distribution function f 2 , corresponding to the difference of the distribution functions between the two species, and the electric field E decay at the faster optimal time rate (1 + t) − 5 4 in L 2 -norm.…”

“…For comparison, we include the known time decay rates of the global solution to the linearized Boltzmann equation (2.18) as follows, ref. [20,25] and references therein. Theorem 5.7.…”

“…In particular, the macroscopic part of the distribution function f 2 decay at exponentially and and microscopic part of the distribution function f 2 decays at the optimal rate (1 + t) − 5 4 (refer to Theorem 2.4 for details). Here we recall that the macroscopic part and microscopic part related to the distribution function f 1 decay at the different optimal rates (1 + t) − 3 4 and (1 + t) − 5 4 respectively as shown in [25]. These optimal algebraic time decay rates also established for the distribution solution (f 1 , f 2 , E, B) to the nonlinear two-species VMB system, and we are able to show that the distribution function f 2 and the electric field E converge to zero state at the faster optimal time rate (1 + t) − 5 4 than the optimal rate (1 + t) − 3 4 for the function f 1 and the electric field B (refer to Theorem 2.5 and Theorem 2.6 for details).…”

Functionalized Benzimidazole Scaffolds: Privileged Heterocycle for Drug Design in Therapeutic Medicine

“…). The interaction of compound 220 with calf thymus DNA demonstrated that this compound could effectively intercalate into DNA to form 220 –DNA complex that might block DNA replication and thereby exert good antimicrobial activity . After oral administration, the onset of the antisecretory effect of omeprazole 9 occurred within 1 h and maximum effect occurred within 2 h. At 24 h, inhibition of secretion was approximately 50% of maximum and duration of inhibition lasted up to 72 h according to pharmacokinetic/pharmacodynamic study .…”

Stability of the Superposition of a Viscous Contact Wave with Two Rarefaction Waves to the Bipolar Vlasov--Poisson--Boltzmann System

“…Indeed, as analyzed in [7,26,27,28], the long time asymptotic behaviors of global solutions to VPB system are rather complicated under the combined effects of the binary collision among the particles of same charge, the electrostatic potential force, and/or the mutual interactions between different charged particles. In particular, it was proved for the unipolar VPB system in [27] that the appearance of electric field affects the spectrum structure of the linearized unipolar VPB system and causes the slower but optimal (compared with the Boltzmann equation) time convergence rates of global solution to the equilibrium state, and there does not exist wave propagation any more, the readers can refer to [7,28,53] and references therein for details.…”