Multiscale analysis for linear first order PDEs. The finite Larmor radius regime (Q2814476)

The author of this paper investigates the Vlasov model containing the following PDEs: NEWLINE\[NEWLINE \partial_tf+v(p)\cdot \nabla_xf+q(E(t,x)+v(p)\wedge B(t,x))\cdot \nabla_pf=0, \;\;(t,x,p)\in\mathbb{R}_{+}\times\mathbb{R}^3, NEWLINE\]NEWLINE where \(m\) is the particle's mass, \(q\) - charge, \(f=f(t,x,p)\) is the particle distribution depending on time \(t\), \(x\) - position, \(p\) - momentum, \(v(p)\) - velocity function. In relativistic models \(v(p)= pm^{-1}(1+|p|^2m^{-2}c_0^{-2})^{-1/2}\), where \(c_0\) is the light speed in the vacuum. If one adds the Maxwell equations, then one obtains the so-called Vlasov-Maxwell system.NEWLINENEWLINEDifferent mathematical problems concerning the dynamical processes in tokamak plasmas, were intensively studied during last decades. This is connected with the so-called ``finite Larmor radius regime'' which can be defined as the author does by: ``the typical length in the perpendicular directions (w.r.t. the magnetic lines) is of the same order as the Larmor radius, and the typical length in the parallel direction is much larger''. In this paper the author investigates asymptotic regimes of the Vlasov equation, that of point of view of Physics could be considered as a linear transport equation. Note that here the self-consistent electromagnetic field can be neglected. It turns out that it is connected with the study of the finite Larmor radius regime. Here the author considers the problem NEWLINE\[NEWLINE \partial_tu^{\varepsilon }+\text{div}_{y}\{u^{\varepsilon }a\}+ \frac{1}{\varepsilon}\text{div}_{y}\{u^{\varepsilon }b\}=0, \;\;(t,y)\in\mathbb{R}_{+}\times\mathbb{R}^m NEWLINE\]NEWLINE under initial condition \(u^{\varepsilon }(0,y)=u^{in}(y)\), \(y\in\mathbb{R}^m\), where the vector fields \(a:\mathbb{R}_{+}\times\mathbb{R}^m\to \mathbb{R}^m\), \(b:\mathbb{R}^m\to \mathbb{R}^m\) (\(a=a(t,y)\), \(b=b(t,y)\)) are smooth and divergence free, \(a\in L_{loc}^{1}(\mathbb{R}_{+};W_{loc}^{1,\infty } (\mathbb{R}^{m}))\), \(b\in W_{loc}^{1,\infty } (\mathbb{R}^{m})\). Assume that certain growth conditions hold: \(\forall T>0 \;\exists C_T>0\) such that \(|a(t,y)|\leq C_T(1+|y|)\), \((t,y)\in [0,T]\times\mathbb{R}^m\), and \(\exists C>0\) such that \(|b(t,y)|\leq C(1+|y|)\), \(y\in\mathbb{R}^m\). Then a proper limit model is constructed, and a convergence process is established under any initial condition. Using the mean ergodic theorem it turns out that one can separate the fast and slow dynamics. This method can be applied to similar problems in lots of dynamical models. Also it is shown that there exist collision operators and then effective diffusion matrices of the limit models can be constructed.