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Regularity and propagation of moments in some nonlinear Vlasov systems. (English) Zbl 0984.35102
The authors consider systems of Vlasov-Poisson type \[ \partial_t f+v\cdot\nabla_x f+\text{div}_v(Ff)=0,\quad \] \[ F(t, x)=\pm\int_{{\mathbb R}^3}|x-y|^{-3}(x-y)\rho(t, y) dy, \quad \] \[ \rho(t, x)=\int_{{\mathbb R}^3}f(t, x, v) dv \] with data \(f(0, x, v)=f^0(x, v)\geq 0\). The principal question addressed is whether the velocity moments \(M_k(t)=\sup_{s\in [0, t]} \int_{{\mathbb R}^6}|v|^k f(s, x, v) dx dv\) propagate, i.e., does a bound on \(M_k(0)\) and \(f^0\) (in an appropriate norm) imply that \(M_k(t)\) can be controlled through \(M_k(0)\) and this norm? This problem is intimately connected with the existence of global solutions to the system, since the relevant terms can be bounded by means of suitably high moments. It is shown in the paper that all moments \(k>2\) propagate, in generalization of earlier work in [P. L. Lions and B. Perthame, Invent. Math. 105, 415-430 (1991; Zbl 0741.35061)], where the same has been proved for certain higher moments. More precisely, for the Vlasov-Poisson system the result is as follows. Let \(k>2\), \(\varepsilon>0\), and \(f^0\in L^\infty({\mathbb R}^6)\) be nonnegative and such that \[ \int_{{\mathbb R}^6}(1+|v|^k+|x|^{1/3+\varepsilon})f^0(x, v) dx dv<\infty, \quad \] \[ \int_{{\mathbb R}^3}f^0(x-vt, v) dv\in L^1_{\text{loc}}([0, \infty[; L^{3(k+3)/(k+6)}({\mathbb R}^3)). \] Then there exists a corresponding global weak solution of the system for which in particular it holds that \(\int_{{\mathbb R}^6}(1+|v|^k+|x|^{1/3+\varepsilon})f(\cdot, x, v) dx dv \in L^\infty(0, T)\), for every \(T>0\).

MSC:
35L60 First-order nonlinear hyperbolic equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D10 Statistical mechanical studies of plasmas
35D05 Existence of generalized solutions of PDE (MSC2000)
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