Singularities and statistics in nonlinear PDE

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DMS Award AbstractAward #: 0202531PI: Constantin, Peter Institution: University of ChicagoProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Singularities and Statistics in Nonlinear Partial Differential EquationsA first goal of this proposal is to study mathematically singularities in fluids and plasmas, and their connection to physical processes. Specific problems include free surface pinch-off in fluids, geometric depletion of singularities in active scalars, incompressible Euler and MHD equations related to vortex and magnetic reconnection. A second goal of the proposal is to study statistics of long time behavior of dissipative systems, such as thermal fluid convection. Specific goals include bounds on heat transport in convection, energy cascades, coarse inertial manifolds, and turbulent front propagation.Fluids and plasmas involve many active length and time scales that interact dynamically. The long time behavior of complex systems, such as turbulent convection, involves also many interacting spatial and temporal scales. Mathematical studies are needed to guide computer simulations. What quantities are well behaved, and can be safely predicted with rather inexpensive, coarse computations? What conditions produce rapid generation of small scales? Which of these are important and result in qualitative changes? What are the characteristics of the ensuing dissipative energy transfer? These are fundamental questions for progress in understanding and computation of complex phenomena. This proposal is aimed at a core of such questions.Date: April 12, 2002