The Vlasov–Maxwell–Boltzmann system with a uniform ionic background near Maxwellians

Abstract

The two-species Vlasov–Maxwell–Boltzmann system is an important model for plasma physics describing the time evolution of dilute charged particles consisting of electrons and ions under the influence of the self-consistent internally generated Lorentz forces. In physical situations the ion mass is usually much larger than the electron mass so that the electrons move much faster than the ions. Thus, the ions are often described by a fixed ion background and only the electrons move. For such a simple case, the two-species Vlasov–Maxwell–Boltzmann system is reduced to the one-species Vlasov–Maxwell–Boltzmann system.Although the one-species Vlasov–Maxwell–Boltzmann system is a simplified model of the two-species Vlasov–Maxwell–Boltzmann system, its global well-posedness theory near a given global Maxwellian in the perturbative framework is more difficult than the two-species case, which is partially due to the slow-decay of the electromagnetic field and up to now, the problem on the construction of global in time solutions near a given global Maxwellian in the perturbative framework for the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions remains unsolved. It is shown in this paper that the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions including the cutoff inverse power law potentials is globally well-posed provided that the perturbative initial data satisfies certain regularity, smallness, and integrability conditions. Our analysis is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the cutoff Boltzmann operator and the other is to design a delicate temporal energy X(t)-norm to obtain its uniform bound. As a by-product of our analysis, we can also deduce certain temporal decay estimates on the global solutions constructed above for the same range of intermolecular collisions. Notice that even for the hard sphere model, although the global solutions have been successfully constructed in [8], its large time behaviors are unknown, cf. [8].