## Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data.(English)Zbl 0810.35089

From the introduction: We study the following system of partial differential equations, which is called the Vlasov-Poisson system ${\partial\over {\partial t}} f(t,x,v)+ v{\partial \over {\partial x}} f(t,x,v)+ E(t,x) {\partial\over {\partial v}} f(t,x,v)=0,\;E(t,x)= \gamma\int_{\mathbb{R}^ n} {{x-y} \over {| x-y|^ n}} \rho(t,y) dy,\tag{1}$ where $$\rho(t,x):= \int_{\mathbb{R}^ n} f(t,x,v)dv$$, $$t\geq 0$$, $$x,v\in \mathbb{R}^ n$$, $$\gamma=\pm 1$$, $$f|_{t=0}= \varphi$$.
We denote this system by (VP). The function $$f$$ describes the development in time of a particle-distribution in the phase-space $$\mathbb{R}^ n \times\mathbb{R}^ n$$. The motion of the particles is, via Vlasov’s equation (1) determined by $$E$$, which is a Newtonian force (attracting) for $$\gamma=-1$$ and a Coulomb force (repulsing) for $$\gamma=1$$. $$E$$ at the time $$t$$ is, via Poisson’s equation, determined by the particle- distribution at $$t$$, so that the problem is nonlinear. The distribution at time 0 is given by $$\varphi$$.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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### References:

 [1] Bardos, C; Degond, P; Bardos, C; Degond, P, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, C.R. acad. sci. Paris Sér. I math., Ann. inst. H. Poincaré anal. non linéaire, 2, 101-118, (1985) · Zbl 0593.35076 [2] Batt, J, Fixpunktprobleme bei partiellen differentialgleichungen im zusammenhang mit dem statistischen anfangswertproblem der stellardynamik, Dissertation, (1962), Aachen [3] Batt, J, Ein existenzbeweis für die Vlasov-gleichung der stellardynamik bei gemittelter dichte, Arch. rational mech. anal., 13, 296-308, (1963) · Zbl 0118.46601 [4] Batt, J, Global symmetric solutions of the initial value problem of stellar dynamics, J. differential equations, 25, 342-364, (1977) · Zbl 0366.35020 [5] Batt, J, Recent developments in the mathematical investigation of the initial value problem of stellar dynamics and plasmaphysics, Ann. nuclear energy, 7, 213-217, (1980) [6] {\scJ. Batt}, Unpublished paper, München, 1989. [7] Dieudonné, J, Treatise on analysis, Vol. II, (1970), New York · Zbl 0202.04901 [8] Hartman, P, Ordinary differential equations, (1964), New York · Zbl 0125.32102 [9] Horst, E, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I, Math. meth. appl. sci., 3, 229-248, (1981) · Zbl 0463.35071 [10] Horst, E, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II, Math. meth. appl. sci., 4, 19-32, (1982) · Zbl 0485.35079 [11] Kurth, R, Das anfangswertproblem der stellardynamik, Z. astrophys., 30, 213-229, (1952) · Zbl 0046.23804 [12] Pfaffelmoser, K, Globale Lösungen des dreidimensionalen Vlasov-Poisson systems, Dissertation, (1989), München · Zbl 0722.35090 [13] Schaeffer, J, Global existence for the Poisson-Vlasov system with nearly symmetric data, J. differential equations, 69, 111-148, (1987) · Zbl 0642.35058 [14] Ukai, T; Okabe, S, On classical solutions in the large in time of two-dimensional Vlasov’s equation, Osaka J. math., 15, 245-261, (1978) · Zbl 0405.35002
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