Mathematical Sciences: Pattern Formation, Turbulence and Singularities in PDEs

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9302013 Newell The investigator and his colleagues undertake projects aimed at gaining a better understanding of turbulence in its many manifestations and at obtaining macroscopic descriptions for patterns that serve to simplify and unify the behavior of classes of pattern forming systems in fluids and optics seemingly unrelated at the microscopic level. In the turbulence studies, the aim is to understand the origins of short-lived coherent structures and their role in causing intermittent and non-Kolmogorov-like behavior. It is important first to gain a clear picture of weak turbulence theories, the reasons for more than one Kolmogorov finite flux solution, and to understand the roles that these solutions play in producing inverse cascades and the roles that inverse cascades may play in originating collapse structures. The principal aim of the studies on pattern formation is to extend the previous far-from-onset theories to include a description of pattern behavior when the local wavenumber is driven outside of a stability band known as the Busse balloon. The results of the research so far have led to a properly regularized theory of patterns that can handle pattern singularities such as dislocations and disclinations. Moreover, the latest work suggests that all point defects of almost periodic patterns are composed of the two elementary disclinations. Patterns of an almost periodic nature arise all over the place. One sees them in the sand ripples under an advancing or receding tide, in geological formations, on the coats of animals, as fingerprints. They occur in systems driven far from equilibrium by some external stress that acts to destabilize spatially and temporally uniform states and preferentially amplify certain shapes and configurations. Geometrical symmetries, such as rotation in the plane, lead to degeneracies (the wavelength of a periodic pattern may be chosen but the direction remains undetermined) and the competition betwee n the various amplified states leads to patterns that often consist of a mosaic of almost periodic patches separated by grain boundaries, domain walls and point defects. The goal of this research is to build a macroscopic field theory for patterns that will unify and simplify the behavior of systems seemingly unrelated at the microscopic level and that can capture both the smooth (field) and singular (particle) components of the pattern.