Abstract
The quasineutral limit of the three dimensional compressible Euler-Poisson (EP) system for ions in plasma under strong magnetic field is rigorously studied. It is proved that as the Debye length and the Larmor radius tend to zero, the solution of the compressible EP system converges strongly to the strong solution of the one-dimensional compressible Euler-equation in the external magnetic field direction. Higher order approximation and convergence rates are also given and detailed studied.
Highlights
We consider the following Euler-Poisson system∂tn + ∇ · = 0, (1a) e×u∂tu + u · ∇u + Ti∇ ln n + ε = −∇φ, (1b) λ2∆φ = eφ − n,(1c) where n(t, x) and u(t, x) are the density and the velocity vector of the ions in a plasma with magnetic field, φ(t, x) is the electric potential at time t > 0 and x ∈ Rd, d ≤ 3
E = (0, 0, 1)T and e/ε is the magnetic field with magnitude 1/ε such that if u = (u1, u2, u3) e × u = (−u2, u1, 0), and λ > 0 is the scaled Debye length, which is a small quantity compared to the characteristic length of physical interest for typical plasma applications
The parameter Ti is the temperature of the ions, which are called cold when Ti = 0
Summary
(1c) where n(t, x) and u(t, x) are the density and the velocity vector of the ions in a plasma with magnetic field, φ(t, x) is the electric potential at time t > 0 and x ∈ Rd, d ≤ 3. E = (0, 0, 1)T and e/ε is the magnetic field with magnitude 1/ε such that if u = (u1, u2, u3) e × u = (−u2, u1, 0), and λ > 0 is the scaled Debye length, which is a small quantity compared to the characteristic length of physical interest for typical plasma applications. The parameters in this equation have obvious physical meanings. Euler-Poisson equation, quasineutral limit, strong magnetic field.
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