We consider pattern forming systems associated with gradient flow structures. Examples include dewetting of surfaces, spinodal decomposition of binary alloys, and swarming in animal populations. Mathematical models for these problems, in the form of nonlinear integrodifferential equations, have solutions with complex dynamics starting from random initial data. We will discuss some recent advancements in this area including rigorous and computational results for coarsening rates of labyrinthian and other pattern types as well as more applied applications such as microfluidic devices and image processing.
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