The purpose of this research will be to analyze models of the time evolution of collisionless plasmas moving under the influence of their self-induced electromagnetic fields. These models take the form of systems of partial differential equations, the Poisson- Vlasov system and the Maxwell-Vlasov system. The unknown in each model is the density of the plasma, and the Maxwell-Vlasov system is somewhat more complete of the two systems, exhibiting electrical and magnetic effects as well as relativistic. Globally symmetric solutions were first established in 1952, and numerous extensions followed. However, the problem of global existence of classical solutions for general data is still open for either system. Recent results establish the breakdown of certain solutions in finite time. One goal of this work will be to show that when the initial data is nearly spherically symmetric, global existence of solutions is guaranteed. A second line of investigation proposes to model the interaction between the solar wind and the earth's magnetic field. By this one understands the neutral stream of charged particles moving rapidly away from the sun. It consists mainly of protons and electrons which, when they encounter the earth's magnetic field, induce currents adding to the total magnetic field. The resulting interaction is believed to account for the aurora borealis and geomagnetic storms. Another feature of this interaction is a bow shock wave across which the behavior of the plasma changes abruptly. The precise nature of shock waves in collisionless plasmas is still unclear. A mixed initial value-boundary value problem is proposed to model the above interaction. Initial work to be done will be in establishing global existence results for solutions and to determine conditions under which solutions tend to a steady state as time increases. Applications to the physical sciences are clear.
