Systems of Nonlinear Partial Differential Equations in Transport Theory

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9321383 Glassey This award supports research on systems of nonlinear partial differential equations and applications to plasma theory. The work involves both analytical and numerical points of view. The emphasis is on collisional effects in problems arising in plasma physics. This involves the modification of the Vlasov- Maxwell systems through the addition of collision operators of Boltzmann- and Landau-type. The major thrust of this effort is dedicated to the study of Landau damping, in which the long time behavior the initial value problem for the linearized Vlasov equation is prescribed. Work has already been done on finding optimal decay rates in the relativistic case, in which the range of validity is restricted by the fact that collisional effects are neglected. Emphasis will now be placed on understanding decay in the nonrelativistic setting. Studies will also be made on the Vlasov-Einstein system to include a fully relativistic model for stellar dynamic problems. The work is important because of its relation to cosmic censorship. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***