Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. (English) Zbl 0593.35076

Bardos, C.; Degond, P.

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 101-118 (1985).

The authors consider the equations \[ \partial u_{\alpha}/\partial t+(v\cdot \nabla_ x)u_{\alpha}+(q_{\alpha}/m_{\alpha})(E\cdot \nabla_ v)u_{\alpha}=\quad 0,\quad 1\leq \alpha \leq N \] with \(v\cdot \nabla_ x=\sum^{3}_{i=1}v_ i(\partial /\partial x_ i)\), \(E\cdot \nabla_ v=\sum^{3}_{i=1}E_ i(\partial /\partial v_ i)\), where E is the electric field related to \(u_{\alpha}\) by the Poisson equation \[ E=-\nabla_ x\phi =4\pi \sum_{\alpha}q_{\alpha}\int_{R^ 3}((x-y)/| x-y|^ 3)(\int_{R^ 3}u\quad_{\alpha}(y,v,t)dv)dy. \] The existence, the uniqueness of a smooth solution \(u_{\alpha}(x,v,t)\) satisfying the initial conditions \(u_{\alpha}(x,v,0)=u_{\alpha,0}(x,v)\) is proved.

Reviewer: I.Bock

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

Vlasov-Poisson equation; Hamiltonian system; estimates; electric field; Poisson equation

References:

[1]	Arsenev, A. A., Global existence of a weak solution of Vlasov’s system of equations, U. S. S. R. comput. Math. and Math. Phys., t. 15, 131-143 (1975)
[2]	Iordanskii, S. V., The Cauchy problem for the kinelic equation of plasma, Amer. Math. Soc. Trans. Ser., 2-35, 351-363 (1964)
[3]	Klainerman, S., Long time behaviour of the solution to non linear equations, Arch. Rat. Mech. Anal., t. 78, 73-98 (1982) · Zbl 0502.35015
[4]	Klainerman, S.; Ponce, G., Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure and Appl. Math., t. 36, 1, 133-141 (1983) · Zbl 0509.35009
[6]	Ukai, S.; Okabe, T., On the classical solution in the large in time of the two dimensional Vlasov equation, Osaka J. of Math., n° 15, 245-261 (1978) · Zbl 0405.35002