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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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##### References:
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