Asymptotic Problems in Parabolic Equations and in Random Transport

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0405152Koralov The project concerns several closely related problems in the theory of parabolic partial differential equations and in random transport. The goal of the project is to investigate the behavior of solutions to the parabolic Anderson problem, the solutions to the equation of the evolution of a magnetic field in a random flow, and a variety of probabilistic aspects of transport phenomena. For the Anderson problem the study focuses on the equations with random time-dependent potential. In the scalar case this problem can be looked upon as a scalar model for the equation of the evolution of the magnetic field in a random flow. In the vector case the Anderson model is related to passive transport by random flows, which is also a subject of the current project. The study of transport phenomena is concerned with the long-time behavior of ensembles of points as well as connected sets under the action of a large class of physically relevant flows. Most of the proposed problems arise naturally in the study of various physical phenomena in meteorology, oceanography, and the theory of turbulence. In particular, when studying passive transport, one assumes that certain properties of the media are known (for example, while the temperatures or velocities on the surface of the ocean can not be measured in every single point exactly, certain statistical information is assumed to be available). The problem then consists of trying to predict the long-time behavior of a passive scalar (such as an oil spill carried by the currents on the surface of the ocean) based on the statistical properties of the underlying media. Several such problems can be formulated in relatively simple terms, yet the solutions are very non-trivial, and at times surprising.