Mathematical Sciences: Studies in Nonlinear Analysis

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Work to be done will focus on several aspects of the behavior of solutions to nonlinear differential and difference equations. It includes studies of the porous medium flow which occurs traditionally in the study of ideal gas flow, but also arises in plasma physics, population dynamics and heat conduction. Particular emphasis will be placed on the corresponding equations in higher space dimensions. The ultimate goal is to determine the regularity class for weak solutions to the equations. In one dimensional flow the optimal global regularity is Lipschitz continuity for the pressure, but for flows in higher dimensions, the optimal result is unknown. A second line of investigation involves the two-phase Stefan problem which seeks to describe the shape of a body immersed in a fluid of the same material (ice in water) as it diffuses. Intuitively the shapes are expected to be self-similar, but this has been shown to be false. Many facts are known about the two- phase problem in one space dimension, but very little is known about the final shape of a disappearing phase in higher dimensions. Some immediate progress can be expected by using results from the one-dimensional case applied to the radially symmetric case in higher dimensions. Work is continuing on studies of the dynamics of circuits involving two current biased Josephson point junctions. The dynamics are described by pairs of differential equations giving the phase difference in the electron wave function across the junctions. Initial work will concentrate on establishing the existence and stability of in-phase rotations - solutions of the systems which agree for all time and are semi-periodic. For large parameter values in the equations, in-phase rotations are well known since the equations reduce to that of a damped pendulum. However, as the parameter values are reduced, numerical simulations suggest areas of instability. Efforts will be made to describe the range of instability.