Recent advances in random data problems for nonlinear dispersive equations (II & III)
Prof. Yu Deng
2023-07-18 11:15-12:15
MCM410
Speaker: Prof. Yu Deng (University of Southern California)Time: 11:15-12:15 July 18, 2023 (Tuesday) & July 20, 2023 (Thursday)
Place: MCM410
Title: Recent advances in random data problems for nonlinear dispersive equations (II & III)
Abstract:
II: We continue with the last lecture and discuss some details in the proof of our main results. In this talk we start with the notion of probabilistic scaling, and then introduce the methods of random averaging operators and random tensors, which can be viewed as the analog of regularity structures in the Schrodinger context. In the Schrodinger context, we will show how these methods allow us to prove almost sure local well-posedness results in the full subcritical range. As a consequence, we also obtain the the invariance of Phi_2^p measures under Schrodinger dynamics.
III: We continue with the last lecture and move on to the invariance of Phi_3^4 measures under the wave dynamics. Compared to the Schrodinger problems, it has the advantage that the wave equation gains one derivative, and the disadvantage that the Gibbs measure is now singular with respect to the underlying Gaussian measure. In view of these differences, the proof will involve new ingredients, including a caloric gauge representation, combinatorics of molecules, and a bilinear random tensor estimate.
