Stochastic Navier-Stokes Equations on Spheres and Spherical Shells

Zdzislaw Brzezniak is currently a Professor in Mathematics at the University of York, U.K. He obtained his PhD in 1988 at Jagellonian University, Poland. Since August 2005 he has been at the University of York as a Professor. His main research areas are stochastic analysis, partial differential equations, stochastic partial differential equations, geometric analysis, Feynman path integrals, and theoretical mathematical finance. He has published over 130 journal articles on many topics and 3 books on stochastic analysis including a very popular undergraduate textbook on Stochastic Calculus published by Springer in 1998.Gaurav Dhariwal is currently a Markets Risk Manager with Lloyds Banking Group (LBG), London, UK. Prior to moving to the LBG, he was a postdoc at the Vienna University of Technology with a research focus on the analysis of Stochastic Partial Differential Equations. In particular, he is interested in the mathematical analysis of models arising in the field of fluid mechanics and population dynamics. He acquired a dual degree of BS and MS in Mathematics and Physics from the Indian Institute of Science Education and Research, Thiruvananthapuram (IISER - TVM) with a distinction and then moved to York, UK where he acquired his PhD in Mathematics under the supervision of Prof. Zdzislaw Brzezniak. Gaurav has been awarded several fellowships and awards; INSPIRE fellowship from the Government of India, MITACS fellowship by the Government of Canada, Anand Ramachandran Memorial Prize for best PhD thesis from the Department of Mathematics, University of York and London Mathematical Society Grant - Research in Pairs are to name a few.Beniamin Goldys is currently a Professor in the School of Mathematics and Statistics at the University of Sydney, Australia. He carries out research in the area of stochastic (ordinary and partial) differential equations and their applications in Financial Mathematics and Science. A major challenge arising in sciences, finance, and biology is to describe and analyse complex systems evolving randomly in space and time. Stochastic (partial) differential equations are one of the main tools to tackle this challenge. It is a fascinating area of mathematical research involving many areas of Pure Mathematics (such as Functional Analysis, Partial Differential Equations, and Ergodic Theory) in order to study complex real-world problems. His specific research areas include stochastic partial differential equations including stochastic geometric partial differential equations, stochastic boundary value problems, stochastic fluid dynamics, ergodic theory of infinite-dimensional diffusions, analysis of infinite-dimensional spaces, interest rate derivatives, credit risk, stochastic volatility, mean field games, applications of stochastic control and Hamilton-Jacobi equations in Finance.Quoc Thong Le Gia is currently an Associate Professor in Applied Mathematics at the University of New South Wales, Sydney, Australia. He obtained his PhD in Mathematics at Texas A & M University, College Station, USA in 2003. His main research areas are approximation theory and numerical methods for partial differential equations. He has contributed to many aspects of approximation methods for partial differential equations, pseudo-differential equations, stochastic partial differential equations, and partial differential equations with random coefficients. He has published over 70 publications on various topics and a book on spherical radial basis functions.Manil T. Mohan is currently an Associate Professor at the Department of Mathematics, Indian Institute of Technology Roorkee (IIT Roorkee). He previously held positions as Visiting Scientist at the Indian Statistical Institute (ISI) in Bangalore, Postdoctoral Fellow at the Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM) funded by the National Board of Higher Mathematics (NBHM), Postdoctoral Fellow at the Air Force Institute of Technology, USA funded by the National Research Council (NRC), and Innovation in Science Pursuit for Inspired Research (INSPIRE) faculty at IIT Roorkee. He obtained his PhD in Mathematics at IISER-TVM, and his B.Sc. and M.Sc. degrees in Mathematics from Kerala University. His areas of interest and research is predominantly in deterministic and stochastic partial differential equations, stochastic analysis, mathematical fluid dynamics, control theory and inverse problems. So far, he has published more than 100 research articles in the areas of his interest. He is currently a member of the Indian National Young Academy of Sciences (INYAS).

• 1. Introduction 2. Preliminaries on probability and stochastic processes 3. Calculus in spherical coordinates 4. Navier-Stokes equations on the unit sphere 5. Stochastic Navier-Stokes equations on spheres 6. Navier-Stokes equations on thin shells 7. Stochastic Navier-Stokes equations on thin shells 8. Existence of martingale solutions to stochastic NSEs in bounded domains