Scaling dynamics and stability in extended physical systems

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PegoDMS-0905723 The focus of this work is on understanding stable dynamicalbehavior in complex nonlinear systems. A variety of phenomenaare investigated in models of strong physical interest,especially: dynamic scaling in models of aggregation andcoarsening (coagulation equations with input, simplified graingrowth models in metallurgy, ballistic aggregation with softcollisions); the robustness of coherent structures in nonlinearfield theories (in particular, spectral stability for exactsolitary water waves); and stability analysis for incompressibleviscous fluids in light of recent progress on boundary conditionsfor pressure (addressing domains with corners, coupling tomagnetism and stress relaxation, and the zero-viscosity limit). Work on dynamic scaling in particular aims to further developideas that connect dynamical systems methods with infinitedivisibility theory in probability. Progress in this program leads to improved mathematicalmethods for understanding how coherent behavior emerges incomplex nonlinear systems. The particular models to be studiedmathematically are of fundamental scientific interest, relevantto materials science at the nanoscale, aerosol physics (models ofsmog, soot, and smoke), population genetics, physical chemistry,and other fields. The work on fluid boundaries should lead toimprovements in the numerical modeling of small-scale flows, withconsequences for studying small flying and swimming objects,blood flow, and engineered microfluidic systems, for example.