# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Arsenev, A. A., Existence in the large of a weak solution of Vlasov’s system of equations, Zh. Vychisl. Mat. Mat. Fiz., 15, 136-147, (1975), 276  Batt, J.; Rein, G., Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math., 313, 6, 411-416, (1991) · Zbl 0741.35058  Brenier, Y., A Vlasov-Poisson type formulation of the Euler equations for perfect incompressible fluids, (1989), Rapport de recherche INRIA  Brenier, Y., Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25, 3-4, 737-754, (2000) · Zbl 0970.35110  Caglioti, E.; Marchioro, C., Bounds on the growth of the velocity support for the solutions of the Vlasov-Poisson equation in a torus, J. Stat. Phys., 100, 3-4, 659-677, (2000) · Zbl 1016.76098  Golse, F.; Saint-Raymond, L., The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13, 5, 661-714, (2003) · Zbl 1053.82032  Grenier, E., Defect measures of the Vlasov-Poisson system in the quasineutral regime, Comm. Partial Differential Equations, 20, 7-8, 1189-1215, (1995) · Zbl 0828.35106  Grenier, E., Oscillations in quasineutral plasmas, Comm. Partial Differential Equations, 21, 3-4, 363-394, (1996) · Zbl 0849.35107  Grenier, E., Limite quasineutre en dimension 1, (Journées Équations aux Dérivées Partielles, Saint-Jean-de-Monts, 1999, (1999), Univ. Nantes Nantes), Exp. No. II, 8 · Zbl 1008.35054  Han-Kwan, D., Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36, 8, 1385-1425, (2011) · Zbl 1228.35251  Han-Kwan, D.; Hauray, M., Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation, Comm. Math. Phys., 334, 2, 1101-1152, (2015) · Zbl 1309.35173  D. Han-Kwan, M. Iacobelli, The quasineutral limit of the Vlasov-Poisson equation in Wasserstein metric, 2014, submitted for publication. · Zbl 1367.35180  Horst, E., On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci., 16, 2, 75-86, (1993) · Zbl 0782.35079  Loeper, G., Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. (9), 86, 1, 68-79, (2006) · Zbl 1111.35045  Masmoudi, N., From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26, 9-10, 1913-1928, (2001) · Zbl 1083.35116  Pallard, C., Large velocities in a collisionless plasma, J. Differential Equations, 252, 3, 2864-2876, (2012) · Zbl 1233.35193  Pfaffelmoser, K., Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95, 2, 281-303, (1992) · Zbl 0810.35089  Schaeffer, J., Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations, 16, 8-9, 1313-1335, (1991) · Zbl 0746.35050  Villani, C., Topics in optimal transportation, Grad. Stud. Math., vol. 58, (2003), American Mathematical Society Providence, RI · Zbl 1106.90001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.