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Blog Post number 1
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publications
Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators
Published in Electronic Journal of Differential Equations (EJDE), 2021
Recommended citation: Usman Hafeez, Theo Lavier, Lucas Williams, Lyudmila Korobenko. (2021) "Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators." EJDE. No. 82, pp. 1-19
talks
Optimal Transport and the Compressible Semi-Geostrophic Equations
Published:
The semi-geostrophic equations for a compressible fluid, first analyzed by Cullen and Maroofi [2003], provide a simplified model of the formation and evolution of atmospheric fronts. I will discuss the use of semi-discrete optimal transport theory to rigorously prove the existence of global-in-time weak solutions as the limit of spatially discrete approximations. This constructive proof of the existence of weak solutions directly extends the work of Bourne et al. [2022] from the incompressible to the compressible setting. I will also show some of the numerical results and discuss the challenges we faced in implementing this numerical method.