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Diffusive Limit of the Vlasov-Poisson-Boltzmann System for the Full Range of Cutoff Potentials
- Publication Year :
- 2023
-
Abstract
- Diffusive limit of the Vlasov-Poisson-Boltzmann system with cutoff soft potentials $-3<\gamma<0$ in the perturbative framework around global Maxwellian still remains open. By introducing a new weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay, we solve this problem for the full range of cutoff potentials $-3<\gamma\leq 1$. The core of this approach lies in the interplay between the velocity weighted $H_{x,v}^2$ energy estimate with time decay and the time-velocity weighted $W_{x,v}^{2,\infty}$ estimate with time decay for the Vlasov-Poisson-Boltzmann system, which leads to the uniform estimate with respect to the Knudsen number $\varepsilon\in (0,1]$ globally in time. As a result, global strong solution is constructed and incompressible Navier-Stokes-Fourier-Poisson limit is rigorously justified for both hard and soft potentials. Meanwhile, this uniform estimate with respect to $\varepsilon\in (0,1]$ also yields optimal $L^2$ time decay rate and $L^\infty$ time decay rate for the Vlasov-Poisson-Boltzmann system and its incompressible Navier-Stokes-Fourier-Poisson limit. This newly introduced weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay is flexible and robust, as it can deal with both optimal time decay problems and hydrodynamic limit problems in a unified framework for the Boltzmann equation as well as the Vlasov-Poisson-Boltzmann system for the full range of cutoff potentials. It is also expected to shed some light on the more challenging hydrodynamic limit of the Landau equation and the Vlasov-Poisson-Landau system.<br />Comment: 59 pages
- Subjects :
- Mathematics - Analysis of PDEs
35Q20, 35Q83
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2307.14088
- Document Type :
- Working Paper