Some Nonlinear PDEs from Material Science, Biological Sciences, and Fluid Mechanics

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Su0708479 The purpose of the project is to study several types ofnonlinear partial differential equations that arise in materialscience, solid mechanics, and cell biology. In particular, thegoals of project are to study the relation between a variety ofplate theories and three-dimensional elasticity theory byderiving the asymptotic limit of three-dimensional nonlinearelasticity, to justify the limit of a singularly perturbedfunctional as an Aviles-Giga functional, to investigate theevolution of interfaces in multi-phase polymer crystals growth byaddressing the stability of moving fronts, and to deepenunderstanding of the homogenization limit of parabolic equationsand of systems of elliptic equations in cell biology and polymersciences. The investigator studies nonlinear pertial differentialequations that describe important behaviors of systems inpolymers, materials, and cell biology. Clear understanding ofthe stability of moving interfaces helps control the interfacebetween the melt and crystal to ensure high quality of polymerproduction. Theoretical guided computation of the Young modulusin polymer nanocomposites is important to measure the strength ofmicro-packaging material for computer chips. Identifying thecorrected diffusion process for protein IP3 from cell membrane tohighly convoluted endoplastic reticulum can save huge amounts ofmoney and time in fundamental scientific research in cellbiology.