Level: BSc, MSci, MSc
Title: Applications of harmonic analysis to nonlinear partial differential equations
Supervisor:

Dr Arick Shao
Research Area: Geometry and Analysis
Description:

In the past few decades of research, the Fourier transform, and more generally the field of harmonic analysis, has emerged as a tremendously valuable tool in the study of nonlinear partial differential equations. This project explores applications of harmonic analysis in the study of nonlinear dispersive PDEs (such as Schrödinger and wave equations). The focus will be on the role of the Fourier transform in establishing existence and uniqueness of solutions, as well as the development of analytical tools that are applied toward solving these equations.

Possible topics include the following:

  • Local-in-time existence and uniqueness of solutions of nonlinear Schrödinger or wave equations;
  • Frequency decompositions and Littlewood-Paley theory;
  • Differential and integral inequalities (Sobolev and product estimates);
  • Strichartz and multilinear estimates for solutions of dispersive PDE.

Further details for the project will depend on the specific background of the student.
Further Reading:
  • T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis. American Mathematical Society, 2006.
Key Modules:

MTH5102 Calculus III

MTH5104 Convergence and Continuity

MTH5105 Differential and Integral Analysis

MTH5123 Differential Equations
Other Information:

While not formally required, any additional background in real analysis or partial differential equations would be helpful.
Current Availability: Yes