Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits (Q2823010)

In this paper, the authors consider the instability of solutions to the relativistic Vlasov-Maxwell system: NEWLINE\[NEWLINE (VM) \;\;\partial_tf+\hat{v}\cdot \nabla_xf+ (E+c^{-1}\hat{v}\times B)\cdot\nabla_vf=0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE c^{-1}\partial_tB+\nabla_x\times E=0, \quad \nabla_x\cdot E=\rho -1, NEWLINE\]NEWLINE NEWLINE\[NEWLINE -c^{-1}\partial_tE+\nabla_x\times B=c^{-1}j, \quad \nabla_x\cdot B=0. NEWLINE\]NEWLINE This system is studied in two limiting frames: (a) the case when the speed of light tends to infinity (clssical case); and (b) the quasineutral limit when the Debye length tends to zero. The above stated system is a part of a mathematical model concerning the evolution of an electron distribution function \(f (t, x, v)\) at time \(t \geq 0\), position \(x \in \mathbb{T}^3\equiv\mathbb{R}^3/\mathbb{Z}^3\), momentum \(v \in\mathbb{R}^3\), and relativistic velocity \(\hat{v}=v(1+|v|^2/c^2)^{-1/2}\). Here, \(c\) is the speed of light, \(\mathbb{T}^3\) is equipped with the normalized Lebesgue measure Leb\((\mathbb{T}^3) = 1\); \(E(x,t)\) and \(B(x,t)\) are three-dimensional vectors, electric and magnetic fields, respectively; \(\rho (x,t)=\int_{\mathbb{R}^3}fdv\) and \(j(x,t)=\int_{\mathbb{R}^3}\hat{v}fdv\) are the charge density and current of electrons, respectively. Here, the authors focus their attention on a strange expression \(c\to +\infty \) (non-relativistic or classical) which sounds incomprehensibly for a strict reader. Then arise the classical Vlasov-Poisson system: NEWLINE\[NEWLINE (VP) \;\;\partial_tf+ v\cdot \nabla_xf+ E\cdot\nabla_vf=0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \nabla_x\times E=0, \quad \nabla_x\cdot E=\rho -1. NEWLINE\]NEWLINE Further, the authors consider the non-dimensional Vlasov-Maxwell system. The regime for this system defined in limit frame \(\alpha =(r_0/\varepsilon_0)^{1/2} \mapsto 1\), \(\varepsilon\to 0\) is characterized with the property that the charge density of ions and electrons are formally equal; \(r_0\) denotes the classical electron radius, and \(\varepsilon_0\) is the vacuum dielectric constant. This regime is known as the quasineutral limit of the Vlasov-Maxwell system. After introducing new variables \((t,x,v)\longmapsto (s,y,v)\equiv (t/\varepsilon ,x/\varepsilon ,v)\), and \(E^{\varepsilon }=\varepsilon E\), \(B^{\varepsilon }=B\), \(f^{\varepsilon }=f\) we obtain an \(\varepsilon \)-system \((VM_{\varepsilon })\) analogical to \((VM)\), and depending continuously on \(\varepsilon \). Thus in the long time, and in the spatially high frequency regime we get that the quasineutral limit leads to the study of the classical limit of the \((VM)\) to the \((VP)\) system. The authors write \(c=1/\varepsilon \), which is too strange. Actually, one may note here that \(1/\varepsilon \) is a very great quantity.NEWLINENEWLINEThe main result is that an instability exists in the classical case. Thus in the classical limit \(\varepsilon\to 0\), with \(\varepsilon = 1/c\) it is constructed a family of solutions that converge initially polynomially very fast to a homogeneous solution of the \((VP)\) system in arbitrarily high Sobolev norm. It is shown that in the classical limit, the \((VM)\) system can develop instability in time \(\log c\), due to the instability of the limiting \((VP)\) system. It is proved that there exists a family of smooth solutions \((f^{\varepsilon } , E^{\varepsilon } , B^{\varepsilon })\) for the \(\varepsilon \)-system \((VM_{\varepsilon })\).NEWLINENEWLINEThe second interesting result is the invalidity of the quasineutral limit in the space \(L^2\) in arbitrarily short time.