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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Gigliola Staffilani (Massachusetts Institute of Technology)

Randomization and nonlinear dispersive equations

I will start by recalling the notion of well-posedness for an initial value problem, I will introduce the Strichartz estimates and indicates some of the proofs one encounters in the deterministic approach. Then I will move to describing the Gaussian measure and the Gibbs measure associated to certain dispersive equations in Hamiltonian form. I will illustrate some of the work of Bourgain in this area and I will show how randomizing the initial data of a Cauchy problem improves well-posedness results. I will conclude with a recent work of myself with Magda Czubak, Dana Mendelson and Andrea Nahmod in which we treat the well-posedness of a geometric wave equation with randomized supercritical data.