Oscillatory Integrals and Partial Differential Equations

Faculty Development Competitive Research Grants Program 2022-2024

The main motivation of this project is to apply techniques from Harmonic Analysis (mainly a delicate treatment of oscillatory integrals) to solve different problem in Partial Differential Equations (PDEs). Our first goal is to make contributions in a long standing conjecture and deduce regularity properties for the solutions of the wave and Schrödinger equations, which are inherit from the behavior of their corresponding initial data. We also want to address the well posedness (existence, uniqueness and stability) of the solutions of various nonlinear dispersive PDEs when their initial data decays fast at infinity, which is a realistic assumption in many applications. Furthermore, we are interested in analyzing a generalization of the classical Maxwell’s equations, which is known to model new materials with exotic electromagnetic properties (Section 2.3). To implement numerically and validate our theoretical results we plan to use, and possibly improve, two novel approaches for solving PDEs: by means of Neural Networks and Quantum Computing