Global existence for the Poisson-Vlasov system with nearly symmetric data (Q1101613)

The author considers the Cauchy problem for the Poisson-Vlasov equation \[ (1)\quad f_ t+v\cdot \nabla_ xf+E_ f(x,t)\cdot \nabla_ vf=0;\quad f(x,v,0)=f_ 0(x,v), \] where \[ E_ f(x,t):=\int \rho_ f(y,t)| x-y|^{-3}(x-y)dy,\quad \rho_ f(x,t):=\int f(x,v,t)dv \] and \(f_ 0\) is given in \(C\) \(1_ 0({\mathbb{R}}\) \(3\times {\mathbb{R}}\) 3). Global existence theorems are known in the literature either if \(f_ 0\) satisfies some symmetry conditions [see \textit{J. Batt}, ibid. 25, 342-364 (1977; Zbl 0366.35020), \textit{S. Wollman}, ibid. 35, 30-35 (1980; Zbl 0414.76094), \textit{E. Horst}, Math. Methods Appl. Sci. 3, 229-248 (1981; Zbl 0463.35071) and 4, 19-32 (1982; Zbl 0485.35079)] or if \(f_ 0\) is sufficiently small in an appropriate sense [see \textit{C. Bardos} and \textit{P. Degond}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 101-118 (1985; Zbl 0593.35076)]. The purpose of the paper is to combine the two aspects, treating the case in which \(f_ 0\) is close to some spherically symmetric function \(g_ 0.\) More precisely, let \(g_ 0\in C\) \(2_ 0({\mathbb{R}}\) \(3\times {\mathbb{R}}\) 3) be a function, depending on \(| x|\), \(| v|\) and \(x\cdot v\), such that \[ \sup t g_ 0\subset \{(x,v): | x| \geq \delta,\quad | v| \geq \delta,\quad x\cdot v\geq \delta | x| | v| \} \] for some \(\delta\in]0,1[\) and let \(R_ 0,U_ 0>0\) be such that \[ \sup t g_ 0\subset \{(x,v): | x| \leq R_ 0,\quad | v| \leq U_ 0\}. \] The main result asserts that there exists \(\epsilon >0\) such that for every \(f_ 0\in C\) \(1_ 0({\mathbb{R}}\) \(3\times {\mathbb{R}}\) 3) satisfying the conditions \[ \sup t f_ 0\subset \{(x,v): \delta \leq | x| \leq R_ 0,\quad \delta \leq | v| \leq U_ 0,\quad x\cdot v\geq \delta | x| | v| \};\quad \| f_ 0-g_ 0\|_{W^{1,\infty}}<\epsilon, \] there exists a global solution f to problem (1). Moreover f decays in a suitable way as \(t\to +\infty\).