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

(this post was also posted here on my new blog address:http://blog.toannguyen.org/ )

Daniel Han-Kwan and I have just submitted a paper entitled: “Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits”, which is also available on arxiv: arXiv:1506.08537. In this paper, we study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit ${\varepsilon \rightarrow 0}$, with ${\varepsilon}$ being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution ${\mu}$ of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from ${\mu}$ in arbitrary negative Sobolev norms within time of order ${|\log \varepsilon|}$. Second, we deduce the invalidity of the quasineutral limit in ${L^2}$ in arbitrarily short time.

\displaystyle \left \{ \begin{aligned} \partial_t f + \hat{v} \cdot \nabla_x f + (E + \frac{1}{c} \hat{v} \times B)\cdot \nabla_v f & =0, \\ \frac{1}{c} \partial_t B + \nabla_x \times E = 0, \qquad \nabla_x \cdot E &= \rho - 1, \\ - \frac{1}{c} \partial_t E + \nabla_x \times B = \frac{1}{c} j , \qquad \nabla_x \cdot B & =0, \end{aligned} \right. \ \ \ \ \ (1)

describing the evolution of the electron distribution function ${f(t,x,v)}$ at time ${t\ge 0}$, position ${x\in \mathbb{T}^3:= \mathbb{R}^3/ \mathbb{Z}^3}$, momentum ${v \in \mathbb{R}^3}$ and relativistic velocity

$\displaystyle \hat{v} = \frac{v}{\sqrt{1+ \frac{|v|^2}{c^2}}}.$

The electric and magnetic fields ${E(x,t), B(x,t)}$ are three-dimensional vector fields, satisfying the classical Maxwell equations, with sources given by

$\displaystyle \rho (t,x)= \int_{\mathbb{R}^3} f \; dv, \quad j (t,x)= \int_{\mathbb{R}^3} \hat{v} f\; dv,$

which denote the usual charge density and current of electrons. The background ions are assumed to be homogeneous with a constant charge density equal to one.

Here the parameter ${c}$ is the speed of light; we focus in the regime where ${c \rightarrow +\infty}$, that is known as the classical limit of the Vlasov-Maxwell system. For each fixed ${c>0}$, the global-in-time Cauchy theory for smooth solutions of (1)remains an outstanding open problem. However, there are local strong solutions and continuation conditions (for instance, Glassey-StraussSingularity formation in a collisionless plasma could occur only at high velocities. Arch. Rational Mech. Anal. 92 (1986), and recently Luk-StrainA new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys. 331 (2014), among others) or global weak solutions (DiPerna-Lions, Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42 (1989)).

Our constructed solutions are sufficiently smooth. The formal limit when ${c\rightarrow +\infty}$ is the classical Vlasov-Poisson system:

\displaystyle \left \{ \begin{aligned} \partial_t f + {v} \cdot \nabla_x f + E \cdot \nabla_v f & =0, \\ \nabla_x \times E = 0, \qquad \nabla_x \cdot E &= \rho - 1. \end{aligned} \right.

This limit was justified on finite intervals of time in the independent and simultaneous works of Asano-Ukai, Degond, and Schaeffer, back in 1986. There appears no further study afterwards in the literature concerning the classical limit problem. What we prove in this paper is that the classical limit is invalid in large times of order ${\log c}$. The result is as follows:

Theorem 1 (Instability in the classical limit) Set ${\varepsilon = \frac 1c}$. There are equilibria ${\mu(v)}$ so that for any ${m,s,s',p>0}$, there exist a family of smooth solutions ${(f_{\varepsilon}, E_{\varepsilon}, B_{\varepsilon})_{\varepsilon>0}}$ of (1), with ${f_\varepsilon\ge 0}$, and a sequence of times ${s_\varepsilon = \mathcal{O}(|\log \varepsilon|)}$ such that

$\displaystyle \| (1+| v|^2)^{\frac m2} ({f^{\varepsilon}}_{\vert_{s=0}}- \mu)\|_{H^s(\mathbb{T}^3\times \mathbb{R}^3)} \leq \varepsilon^p, \ \ \ \ \ (2)$

but

$\displaystyle \liminf_{\varepsilon \rightarrow 0} \left\| f^\varepsilon(s_\varepsilon) - \mu\right\|_{H^{-s'}(\mathbb{T}^3\times \mathbb{R}^3)} >0, \ \ \ \ \ (3)$

$\displaystyle \liminf_{\varepsilon \rightarrow 0}\left\| \rho^\varepsilon(s_\varepsilon) - 1 \right\|_{H^{-s'}(\mathbb{T}^3)} >0, \quad \liminf_{\varepsilon \rightarrow 0} \left\| j^\varepsilon(s_\varepsilon) \right\|_{H^{-s'}(\mathbb{T}^3)} >0, \ \ \ \ \ (4)$

$\displaystyle \liminf_{\varepsilon \rightarrow 0} \left\| E^\varepsilon(s_\varepsilon) \right\|_{L^2(\mathbb{T}^3)} >0. \ \ \ \ \ (5)$

Next, in this paper, we are interested in justifying the quasineutral limit, as ${\varepsilon \rightarrow 0}$, of the Vlasov-Maxwell system:

\displaystyle \left \{ \begin{aligned} \partial_t f + \hat{v} \cdot \nabla_x f + (E + \alpha \hat{v} \times B)\cdot \nabla_v f & =0, \\ \alpha \partial_t B + \nabla_x \times E = 0, \qquad \varepsilon^2 \nabla_x \cdot E &= \rho - 1, \\ - \alpha \varepsilon^2 \partial_t E + \nabla_x \times B = \alpha j, \qquad \nabla_x \cdot B & =0, \end{aligned} \right. \ \ \ \ \ (2)

with

$\displaystyle \hat{v} = \frac{v}{\sqrt{1+ \alpha{\varepsilon^2 |v|^2}}}.$

In physical units, the parameters ${\alpha, \varepsilon}$ are given by

$\displaystyle \alpha = \sqrt{\frac{r_0}{\varepsilon_0}} , \qquad \varepsilon = \sqrt{\frac{\varepsilon_0}{r_0 c^2}},$

where ${r_0}$ denotes the classical electron radius, ${\varepsilon_0}$ is the vacuum dielectric constant, and ${c}$ is still the speed of light. The parameter ${\varepsilon}$ corresponds to the classical Debye length of the electrons; see Puel and Saint-Raymond, Quasineutral limit for the relativistic Vlasov-Maxwell system. Asymptot. Anal. 40 (2004), for further discussions. In this work, we are interested in the regime where ${\alpha \sim 1}$ and ${\varepsilon \rightarrow 0}$, a limit in which the charge density of ions and electrons are formally equal. We shall thus refer to this problem as thequasineutral limit of the Vlasov-Maxwell system. Note that ${\alpha}$ is equal to the ratio between ${\frac{1}{\varepsilon}}$ and ${c}$, so that this means that we consider that the inverse of the Debye length is of the same order as the speed of light. For simplicity, throughout the paper, we set ${\alpha=1}$.

The quasineutral limit of the Vlasov-Maxwell system has been studied previously by Brenier, Mauser and Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system. Commun. Math. Sci. 1 (2003), and Puel and Saint-Raymond, Quasineutral limit for the relativistic Vlasov-Maxwell system. Asymptot. Anal. 40 (2004), in the case where the initial density distribution converges in some weak sense to a monokinetic distribution (that is, a Dirac delta function in velocity). The convergence to monokinetic distributions can be interpreted from the physical point of view as vanishing temperature: therefore, this is sometimes referred to as the cold electrons limit.

Formally, in the limit ${\varepsilon \rightarrow 0}$, it is straightforward to obtain the expected formal limit, a system we shall call the kinetic eMHD system:

\displaystyle \left \{ \begin{aligned} \partial_t f^0 + v \cdot \nabla_x f^0 + (E^0 + v \times B^0)\cdot \nabla_v f^0 & =0, \\ \partial_t B^0 + \nabla_x \times E^0 = 0, \qquad \rho^0 &= 1, \\ \nabla_x \times B^0 = j^0, \qquad \nabla_x \cdot B^0 & =0. \end{aligned} \right. \ \ \ \ \ (7)

By imposing that the distribution function is monokinetic, that is considering the ansatz ${f^0(t,x,v) =\rho^0(t,x) \delta_{v=u^0(t,x)}}$, where ${\delta}$ stands for the Dirac measure, it follows that ${f^0}$ is a solution in the sense of distribution to (7) if and only if ${(\rho^0,u^0)}$ satisfies the following hydrodynamic equations:

\displaystyle \left \{ \begin{aligned} \partial_t u^0 + \nabla \cdot (u^0 \otimes u^0) =E^0 + u^0 \times B^0, \\ \partial_t B^0 + \nabla \times E^0 = 0, \qquad \rho^0 &= 1, \\ \nabla \times B^0 = j^0, \qquad \nabla \cdot B^0 & =0. \end{aligned} \right. \ \ \ \ \ (8)

This system is known in the literature as the electron Magneto-Hydro-Dynamics equations (eMHD). This motivates the choice of the name kinetic eMHD for~(7). Brenier, Mauser and Puel, and Puel and Saint-Raymond justify the eMHD system from (6) in the above monokinetic situation via the so-called modulated energy (or relative entropy) method devised earlier by Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25 (2000). In this paper, we rather focus on the question of validity of (7) in the quasineutral limit.

The validity of quasi-neutral limit of Vlasov-Poisson has been addressed and settled recently by D. Han-Kwan and his collaborators. One purpose of this work is to extend the instability results for Vlasov-Poisson to Vlasov-Maxwell systems. Our second main result reads as follows.

Theorem 2 (Invalidity of the quasineutral limit) There are equilibria ${\mu(v)}$ so that for any ${m,s,p>0}$, there exist a family of smooth solutions ${(f_{\varepsilon}, E_{\varepsilon}, B_{\varepsilon})_{\varepsilon>0}}$ of (6), with ${f_\varepsilon\ge 0}$ and a sequence of times ${t_\varepsilon = \mathcal{O}(\varepsilon |\log \varepsilon|)\rightarrow 0}$, such that

$\displaystyle \| (1+| v|^2)^{\frac m2}({f_{\varepsilon}}_{\vert_{t=0}}- \mu)\|_{H^s(\mathbb{T}^3\times \mathbb{R}^3)} \leq \varepsilon^p, \ \ \ \ \ (9)$

but

$\displaystyle \liminf_{\varepsilon \rightarrow 0} \left\| f_\varepsilon(t_\varepsilon) - \mu\right\|_{L^2(\mathbb{T}^3\times \mathbb{R}^3)} >0, \ \ \ \ \ (10)$

$\displaystyle \liminf_{\varepsilon \rightarrow 0} \left\| \rho_\varepsilon(t_\varepsilon) - 1 \right\|_{L^2(\mathbb{T}^3)} >0, \quad \liminf_{\varepsilon \rightarrow 0} \left\| j_\varepsilon(t_\varepsilon) \right\|_{L^2(\mathbb{T}^3)} >0, \ \ \ \ \ (11)$

$\displaystyle \liminf_{\varepsilon \rightarrow 0} \varepsilon \left\| E_\varepsilon(t_\varepsilon) \right\|_{L^2(\mathbb{T}^3)} >0. \ \ \ \ \ (12)$

One Response

1. […] This is proved in Han-Kwan & Toan ’16. […]