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Probabilistic methods for nonlinear dispersive PDEs
Dana Mendelson - University of Chicago (Host: Fauci, Lisa and Glatt-Holtz)
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Nonlinear dispersive equations model wave propagation phenomena for many physical systems, from water waves to quantum gases. For the last few decades, research on these equations has centered around questions on the existence of solutions, their long time behavior, and the possibility of singularity formation. Fundamental progress has been made in many settings, yet in some regimes, the nonlinear interactions overwhelm the dispersion of the waves, and standard methods break down.
In recent years, probabilistic tools have been instrumental in analyzing the behavior of these equations in previously inaccessible regimes. This approach, which goes back to the seminal work of Bourgain in the 90s on invariant Gibbs measures for Hamiltonian PDEs, has opened the door to new and exciting questions in a variety of settings. In this talk, we will give a general overview of the progress in this area and discuss some of the topics of current research.
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