On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains (Q1946319)

The global existence problem of the solution of the coupled Vlasov-Poisson (VP) system of plasma physics is analyzed for a finite convex bounded domain. The boundary condition for the electric potential (Dirichlet condition D) and the specular reflexion for the distribution are chosen. The authors discuss the importance and complicated structure of the boundary conditions because of the onset of boundary layer effects. Finally, the global existence theorem for the VP system with the specular reflexion for the distribution function and D for the electric potential is proven. The demonstration is based on previous results of the first and the third author [J. Differ. Equations 247, No. 6, 1915--1948 (2009; Zbl 1181.35291); Arch. Ration. Mech. Anal. 195, No. 3, 763--796 (2010; Zbl 1218.35235)] and uses refined boundary estimates and constructs relevant supersolutions.