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Quasineutral limit for Vlasov-Poisson via Wasserstein stability estimates in higher dimension. (English) Zbl 1375.35555
This interesting paper is devoted to a study of the Vlasov-Poisson system in two and three dimensions which can be seen in some mathematical models in Plasma Physics and other applications. Adding a subscript to each quantity in this system to emphasize the dependence on some parameter \(\varepsilon \) one may construct a similar system, called rescaled Vlasov-Poisson system. It contains partial differential equations (PDE) written in the form \[ \begin{aligned} \partial_tf_{\varepsilon }+v\cdot \nabla_xf_{\varepsilon }+ E_{\varepsilon }\cdot\nabla_vf_{\varepsilon } = 0, \\ E_{\varepsilon } = - \nabla_xU_{\varepsilon }, \\ - \varepsilon^2 \triangle_xU_{\varepsilon } = \int\limits_{\mathbb{R}^d}f_{\varepsilon }dv - \int\limits_{\mathbb{T}^d\times\mathbb{R}^d} f_{\varepsilon }dvdx,\\ f_{\varepsilon }|_{t=0}=f_{0,\varepsilon }\geq 0, \;\int\limits_{\mathbb{T}^d\times\mathbb{R}^d} f_{0,\varepsilon }dxdv = 1, \end{aligned}\tag{1} \] where \(f_{\varepsilon }(t,x,v)\) is the distribution function of the electrons for \(t\in\mathbb{R}^{+}\), \((x, v) \in \mathbb{T}^{d}\times\mathbb{R}^{d}\), \(\mathbb{T}^d\) is the \(d\)-dimensional torus with \(d = 2, 3\). The quantity \(f_{\varepsilon}(t, x, v)dxdv\) is the probability of finding particles with position and velocity close to the point \((x, v)\) in the phase space at time \(t\). The electric potential and the associated electric field are denoted by \(U (t, x)\) and \(E(t, x)\), respectively. Here \(\varepsilon \) is a positive parameter defined as the ratio of the Debye length of the plasma to the size of the domain.
Next, it is defined a Polish space \(({\mathcal{M}},d)\), and \({\mathcal{P}}_p\) (\(p\in [1,\infty )\)) the collection of all probability measures on \(\mathcal{M}\) with finite moment \(p\). Then the \(p\)-Wasserstein distance between two probability measures \(\mu \) and \(\nu \) in \({\mathcal{P}}_p ({\mathcal{M}})\) can be denoted by \(W_p(\mu ,\nu )\). Introduce the Wasserstein space \({\mathcal{P}}_2 ({\mathcal{M}})\) which is the space of probability measures having a finite moment of order 2 and it will always be equipped with the quadratic Wasserstein distance \(W_2\).
The above stated system (1) possesses some energy given by the equality: \[ {\mathcal{E}} (f_{\varepsilon }(t)) = (1/2)\int\limits_{\mathbb{T}^d\times\mathbb{R}^d} f_{\varepsilon }|v|^2dvdx + (\varepsilon^2/2) \int\limits_{\mathbb{T}^d} |\nabla_xU_{\varepsilon }|^2dx . \] Here the authors discuss the behavior of the rescaled Vlasov-Poisson system (1) in asymptotic frame, i.e., there exists some limit (quasineutral limit) as \(\varepsilon\to 0\). It is observed that if \(f_{\varepsilon }\to f\) and \(U_{\varepsilon }\to U\) in (1) as \(\varepsilon\to 0\) (\(f,U\) some limit functions), then the limit of the system under consideration is \[ \begin{aligned} \partial_tf+v\cdot \nabla_xf+ E\cdot\nabla_vf = 0, \\ E = - \nabla_xU, \\ \int\limits_{\mathbb{R}^d}fdv = 1, \\ f|_{t=0}=f_{0}\geq 0, \;\;\int\limits_{\mathbb{T}^d\times\mathbb{R}^d} f_{0}dxdv = 1, \end{aligned}\tag{2} \] and the total energy of the system reduces to the kinetic part of \({\mathcal{E}}(f(t))= (1/2)\int\limits_{\mathbb{T}^{d}\times\mathbb{R}^{d}} f|v|^2dvdx\).
It is assumed that there is a function \(\varphi : (0, 1]\to \mathbb{R}^{+}\) with \(\text{lim}_{\varepsilon\to 0} \varphi (\varepsilon ) = 0\) and constants \(\gamma \), \(\delta_0 \), \(C_0>0\). The authors consider a sequence \(\{f_{0,\varepsilon }\}\) of non-negative initial data in \(L^1\) for the above stated system (1) such that \(\forall \varepsilon \in (0, 1)\) the following assumptions hold: (i) \(\|f_{0,\varepsilon }\|_{\infty }\leq C_0\), \({\mathcal{E}}(f_{0,\varepsilon })\leq C_0\); (ii) The initial data are compactly supported in velocity with \(f_{0,\varepsilon }(x,v)=0\), \(|v|>1/\varepsilon^{\gamma }\); (iii) It is assumed the existence of the decomposition \(f_{0,\varepsilon }=g_{0,\varepsilon }+ h_{0,\varepsilon }\), where \(\{g_{0,\varepsilon }\}\) is a sequence of continuous functions with uniform compact support in \(v\) satisfying some rate estimate and admitting a limit \(g_0\) in the sense of distributions; \(\{h_{0,\varepsilon} \}\) is a sequence of functions satisfying \(W_2(f_{0,\varepsilon },g_{0,\varepsilon })= \varphi (\varepsilon)\) for all \(\varepsilon > 0\).
Assume that \(f_{\varepsilon } (t)\) for all \(\varepsilon\in (0, 1)\) is a global weak solution of the Vlasov-Poisson system with initial condition \(f_{0,\varepsilon }\) in the sense of A. A. Arsen’ev [Zh. Vychisl. Mat. Mat. Fiz. 15, 136–147 (1975; Zbl 0317.35032)]. A filtered distribution function \(\tilde{f}_{\varepsilon }(t,x,v)\) is defined as well.
Under above stated requirements (i)–(iii) it is proved the interesting result that there exist \(T>0\) and \(g(t)\) which is a weak solution of the limit system (2) on the closed interval \([0, T ]\) with initial condition \(g_0\) such that \(\lim\limits_{\varepsilon\to 0}\sup\limits_{t\in[0,T]} W_1(\tilde{f}_{\varepsilon }(t),g(t))=0\).

MSC:
35Q83 Vlasov equations
35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
35D30 Weak solutions to PDEs
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