Abstract
The Vlasov–Poisson–Boltzmann System governs the time evolution of the distribution function for dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over \({\mathbb R^3}\). It is shown that the electric field, which is indeed responsible for the lowest-order part in the energy space, reduces the speed of convergence, hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces; the exact L 2-rate for the former is (1 + t)−1/4 while it is (1 + t)−3/4 for the latter. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.
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Duan, R., Strain, R.M. Optimal Time Decay of the Vlasov–Poisson–Boltzmann System in \({\mathbb R^3}\) . Arch Rational Mech Anal 199, 291–328 (2011). https://doi.org/10.1007/s00205-010-0318-6
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DOI: https://doi.org/10.1007/s00205-010-0318-6