Mathematical Sciences: Nonlinear Partial Differenial Equations

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9622305 Phillips This project is a study of differential equations from two areas, the Ginzburg--Landau equations as they apply to superconductivity and the calculus of variations related to nonlinear elasticity. Features of solutions to the Ginzburg--Landau equations are to be studied. Experimental evidence for the materials they model and numerical simulations for the equations themselves indicate that stable solutions contain coherent pattern formations (vortex arrays). It is proposed to analyze solutions for simple geometries (e.g. cylindrical rods and thin films). The principal investigator seeks to establish the existence of these patterns and to understand the mechanisms that bring them about. He also seeks to determine how one can control the patterns (pin the vortex distributions). Doing this will lead to a better understanding of the models and to estimates for the effectiveness of a material to carry a supercurrent when subjected to an applied current or a magnetic field. The second part of the proposal deals with the regularity of solutions to variational problems related to nonlinear elasticity. P. Bauman and the principal investigator have established that solutions to certain boundary value problems from two dimensional elasticity are locally Lipschitz continuous homeomorphisms. It is proposed to investigate whether or not these solutions have singularities. Analytically this is to ask if the solutions are differentiable everywhere. %%% Scientists describe how a given material can conduct an electric current or how an elastic body can be twisted and bent by expressing such phenomena as solutions to mathematical equations (called partial differential equations) based on the underlying physics in each instance. The principal investigator studies the features of these solutions. For example in the case of a superconducting material (a material that can conduct electricity very efficiently) it is observ ed that the current naturally circulates around a symmetric array of points called vorticies. It is also observed that if the current is sufficiently strong the array begins to drift causing the current to die off and superconductivity is lost. Pattern formation and the onset of instability are features that can be investigated through the solutions. Understanding and being able to predict these features is important both for theory and engineering applications. ***