On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields
The next problem is considered \[ \partial_t \mathcal{F} + V\cdot\nabla_X\mathcal{F}+\frac{q}{m}(V\wedge B^{ext})\cdot\nabla_V \mathcal{F}=Q(\mathcal{F},\mathcal{F}),\,\,\, (t,X,V)\in \mathbb{R}_+\times\mathbb{R}^3\times\mathbb{R}^3, \] \[ \mathcal{F}(0,X,V)=\mathcal{F}^{in}(X,V), \,\,\, (X,V)\in \mathbb{R}^3\times\mathbb{R}^3. \] This model was designed for beams which possess an optical axis, assuming that the particles remain close to the optical axis, having about the same kinetic energy. This paper is devoted to the study of the confinement properties of charged particle beams, under the action of strong magnetic fields parallel to the optical axis. The self-consistent electro-magnetic field is neglected but it is taken into account the collisions between particles. The function \(\mathcal{F}= \mathcal{F}(t,X, V)\geqslant 0\) is the presence density of the charged particles in the position-velocity phase space \((X, V).\) The collisions between the particles are taken into account through the Boltzmann kernel: \[ \begin{multlined} Q(\mathcal{F},\mathcal{F})=\int_{S^2}\int_\mathbb{R}^3\sigma(V-V',\omega)\{\mathcal{F}(V-\omega\otimes\omega(V-V'))\mathcal{F}(V'+\omega\otimes\omega(V-V'))\\ -\mathcal{F}(V)\mathcal{F}(V')\}d V'd\omega.\end{multlined} \] The density is searched in the form \[ \mathcal{F}^\varepsilon=\frac{1}{\varepsilon^3}f^\varepsilon(t,\frac{X_1}{\varepsilon^2}, \frac{X_2}{\varepsilon^2},X_3,\frac{V_1}{\varepsilon},\frac{V_2}{\varepsilon},\frac{V_3-u_3}{\varepsilon}). \] by using a formal expansion \(f^\varepsilon=f+\varepsilon f^1+\varepsilon\varepsilon^2 f^2+\dots\) The main goal of this paper is to derive the expression of the averaged version of the Boltzmann collision operator. It is allows to express the equilibria of the averaged Boltzmann collision kernel in terms of six moments.
scientific article; zbMATH DE number 6388088
scientific article