The focus of this work is on understanding stable dynamical behavior in complex nonlinear systems. A variety of phenomena are investigated in models of strong physical interest, especially: dynamic scaling in models of aggregation and coarsening (coagulation equations with input, simplified grain growth models in metallurgy, ballistic aggregation with soft collisions); the robustness of coherent structures in nonlinear field theories (in particular, spectral stability for exact solitary water waves); and stability analysis for incompressible viscous fluids in light of recent progress on boundary conditions for pressure (addressing domains with corners, coupling to magnetism and stress relaxation, and the zero-viscosity limit). Work on dynamic scaling in particular aims to further develop ideas that connect dynamical systems methods with infinite divisibility theory in probability.
Progress in this program leads to improved mathematical methods for understanding how coherent behavior emerges in complex nonlinear systems. The particular models to be studied mathematically are of fundamental scientific interest, relevant to materials science at the nanoscale, aerosol physics (models of smog, soot, and smoke), population genetics, physical chemistry, and other fields. The work on fluid boundaries should lead to improvements in the numerical modeling of small-scale flows, with consequences for studying small flying and swimming objects, blood flow, and engineered microfluidic systems, for example.
