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Nonlinear and nonlocal partial differential equations


School of Mathematics

Applications accepted all year round Competition Funded PhD Project (European/UK Students Only)

About the Project

PhD Project: nonlinear and nonlocal partial differential equations

Project description. Complex systems in nature and in applied sciences are often described by nonlinear and nonlocal partial differential equations (PDEs). A typical example is the (spatial and/or time) fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion [ACV16, HIK+18]. A mathematical analysis of such an equation is pivotal for the understanding and control of physical systems. However, it is often challenging due to the involved nonlinearity and nonlocality. Variational methods, in particular Wasserstein gradient flow structures [JKO98, Vil03] has been proven to be a powerful and versatile tool for the analysis of nonlinear and nonlocal PDEs.

The aim of this PhD project is to develop Wasserstein gradient flow-type formulations for nonlinear and nonlocal partial differential equations. The key challenge would be to build up new optimal transportation cost functionals and/or new approximation schemes.

The project will be based at the School of Mathematics at University of Birmingham and will be supervised by Dr. Hong Duong (possibly with another member staff). The School of Mathematics at the University of Birmingham is an internationally leading centre for mathematical research, with particular strengths in mathematical analysis, biological mathematics, combinatorics and optimization. The PhD candidate will have many opportunities to interact with leading scientists at the school of mathematics and other departments.

Recent collaborative works of Dr. Hong Duong in the direction of research of the project include [DLPS17, DT18, DLP+18, DL19, DJ19].

Applications before 31 January is strongly encouraged. Please contact for any questions.

References

[ACV16] M. Allen, L. Caffarelli, and A. Vasseur. A parabolic problem with a fractional time derivative. Archive for Rational Mechanics and Analysis, 221(2):603-630, 2016.

[DJ19] M. H. Duong and B. Jin. Wasserstein gradient ow formulation of the time-fractional Fokker-Planck equation. arXiv:1908.09055, 2019.

[DLP+18] M. H. Duong, A. Lamacz, M. A. Peletier, A. Schlichting, and U. Sharma. Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics. Nonlinearity, 31(10):4517-4566, 2018.

[DLPS17] M. H. Duong, A. Lamacz, M. A. Peletier, and U. Sharma. Variational approach to coarse-graining of generalized gradient flows. Calculus of Variations and Partial Differential Equations, 56(4):100, 2017.

[DT18] M. H. Duong and M. H. Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 38:3407, 2018.

[DL19] M. H.Duong and Y. Lu An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 39:5707, 2019.

[HIK+18] M. Hairer, G. Iyer, L. Koralov, A. Novikov, and Z. Pajor-Gyulai. A fractional kinetic process describing the intermediate time behaviour of cellular flows. Ann. Probab., 46(2):897-955, 2018.

[JKO98] R. Jordan, D. Kinderlehrer, and F. Otto. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal., 29(1):1-17, 1998.

[Vil03] C. Villani. Topics in Optimal Transportation. AMS, Providence, RI, 2003.

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;


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