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Random dispersive and wave equations


   School of Mathematics

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  Dr Yuzhao Wang  Applications accepted all year round  Competition Funded PhD Project (Students Worldwide)

About the Project

The central theme of this project is the mathematical analysis of nonlinear dispersive partial differential equations (PDEs), such as the nonlinear Schrödinger equations (NLS) and the nonlinear wave equations (NLW). One of the most fundamental questions is to understand how initial data are propagated by these nonlinear evolution equations. As such, it is of central importance to study the well-posedness (existence, uniqueness and stability under perturbation) of these nonlinear evolution equations. Over the last thirty years, the field of nonlinear dispersive PDEs has seen significant development in the theoretical understanding of some of these fundamental questions and, in particular, multilinear harmonic analysis has played a fundamental role in this development, led by Kenig, Bourgain and Tao among others. Furthermore, in recent years, a combination of deterministic analysis with probability theory has played an important role in the field. Not only did this probabilistic perspective allow us to go beyond the limit of deterministic analysis, but it is in fact important to understand the effect of stochastic perturbation in practice since such stochastic perturbation is ubiquitous. This led to the study of nonlinear dispersive PDEs with rough random initial data and/or singular stochastic forcing. Since the seminal works [1, 2] by Bourgain, this development has been led by Tzvetkov and Burq (both France) and Oh (Edinburgh) and has been increasing more popular.

Stochastic dispersive PDEs appear as “canonical" stochastic quantisation equations, realising certain measures supported on the space of distributions as invariant measures for dynamics. In recent years, we have seen significant progress in the field of singular stochastic parabolic PDEs, led by Hairer [3] and Gubinelli [4] with their collaborators, advancing the understanding of stochastic quantisation equations in the parabolic setting. On the other hand, our understanding in the dispersive setting is far from being satisfactory, compared to the parabolic setting.

The primary purpose of this proposal is to develop novel ideas and tools to further promote our understanding of the dynamics of random (either with random data or driven by stochastic forcing) dispersive PDEs, by using tools from harmonic analysis, PDE techniques (including elliptic and parabolic theory), stochastic analysis and probability theory.

We are looking for an enthusiastic and highly-motivated graduate with 

- a first-class degree in Mathematics or a closely related discipline with a strong mathematical component (Master’s level or equivalent); 

- a solid background in analysis/partial differential equations; 

- excellent programming skills; 

- good communication skills (oral and written). 

Good knowledge of harmonic analysis and stochastic analysis will be advantageous. (See my webpage at https://web.mat.bham.ac.uk/Y.Wang/)

The application procedure and the deadlines for scholarship applications are advertised at https://www.birmingham.ac.uk/schools/mathematics/phd/phd.aspx 

Informal inquiries should be directed to Dr Yuzhao Wang, email: y.wang.14 at bham.ac.uk 

Funding Notes for Chinese candidates: 

The China Scholarship Council (CSC) Scholarship: https://www.csc.edu.cn/chuguo 

China Scholarship Council (CSC) PhD Scholarships Programme at the University of Birmingham: https://www.birmingham.ac.uk/postgraduate/funding/china-scholarship-council-%E2%80%93-university-of-birmingham-phd-scholarships.aspx 

PhD Placements and Supervisor Mobility Grants China-UK: https://www.britishcouncil.cn/en/programmes/education/higher/opportunities/phd


Funding Notes

For UK and EU candidates:
Funding may be available through a college or EPSRC scholarship in competition with all other PhD applications; The scholarship will cover tuition fees, training support, and a stipend at standard rates for 3-3.5 years.
For non-UK/non-EU candidates:
Strong self-funded applicants will be considered; Exceptionally strong candidates in this category may be awarded a tuition fee waiver (for up to 3 years) in competition with all other PhD applications.
Early applications are strongly recommended; the deadline for scholarship applications is midday UK time on 31st January (annually). Strong candidates are encouraged to make an informal inquiry.

References

[1] J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), 1–26.
[2] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445.
[3] M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), no. 2, 269–504.
[4] M. Gubinelli, P. Imkeller, N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6.
[5] https://web.mat.bham.ac.uk/Y.Wang/

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