Global solutions and time decay of the non-cutoff Vlasov-Maxwell-Boltzmann system in the whole space
The authors prove the existence of a unique global in-time and classical solution to a Cauchy problem associated to a Vlasov-Maxwell-Boltzmann system. They start from the problem \(\partial_tF_++v\cdot\nabla_xF_++E+v\times B)\cdot\nabla_vF_+=Q(F_+,F_+)+Q(F_+,F_-)\), \(\partial_tF_-+v\cdot\nabla_xF_--(E+v\times B)\cdot\nabla_vF_-=Q(F_-,F_+)+Q(F_-,F_-)\), \(\partial_tE-\nabla_xB=-\int_{\mathbb R^3}v(F_+-F_-)dv\), \(\partial_tB+\nabla_x\times E=0\), \(\nabla_x\times E=\int_{\mathbb R^3}(F_+-F_-)dv\), \(\nabla_x\cdot B=0\), posed in \(\mathbb R^3\times (0,\infty)\). Here \( F_\pm\) are the number density functions for the ions \((+)\) and the electrons \((-)\) at position \(x\), with velocity \(v\) and at time \(t\). \(E\) and \( B\) are the electric and magnetic fields, respectively. \(Q\) is the Boltzmann collision operator which involves a Boltzmann collision kernel on which the authors impose a special structure. The initial conditions \(F_\pm(0,x,v)=F_{0,\pm}(x,v)\), \(E(0,x)=E_0(x)\) and \(B(0,x)=B_0(x)\) are imposed with initial data satisfying compatibility conditions. The main result proves the existence of a unique global solution under further hypotheses on the data and on the initial data. But the authors claim that these hypotheses are weaker than that imposed by \textit{R. Duan} et al. in [Kinet. Relat. Models 6, No. 1, 159--204 (2013; Zbl 1288.35476)]. The second main result proves that this solution satisfies time decay estimates like \(\mathcal E_{N_0}^k(t)\leq C(1+t)^{-(\rho+k)}\), with \(\mathcal E_{N_0}^k(t)\sim\sum_{\left|\alpha\right|=k}^{N_0}\left\|\partial^\alpha(f,E,B)\right\|\). For the proofs, the authors introduce an exponential time-velocity weight and they use Duhamel's principle. They also use interpolation techniques between negative Sobolev norms.
scientific article; zbMATH DE number 6537584
scientific article