Yiran Hu

Yiran Hu

Ph.D. Candidate in Math@ UT Austin

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Posts

Weak Solutions

less than 1 minute read

Published:

This is a lecture note for a graduate topic course in 2020 Spring on Weak Solutions. Instructor: Philip Isett. Scribe: Yiran Hu.

publications

Global in Time Weak Solutions to Singular 3D Quasi-Geostrophic Systems

Submitted in , 2023

Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These systems describe tratified flows in the atmosphere on a large time scale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in R+×Ω with an equation on the boundary satisfied by the trace, where Ω is either 2D torus or a bounded convex domain in R2. In this paper, we show the existence of global in time weak solutions to a family of singular 3D quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The proof is based on the construction of approximated models which combine the Galerkin method at the boundary and regularization processes in the bulk of the domain. The main difficulty is handling the degeneration of the background density profile at the boundary. Subjects: Analysis of PDEs (math.AP)

Recommended citation: Hu, Yiran. "Global in Time Weak Solutions to Singular 3D Quasi-Geostrophic Systems." arXiv preprint arXiv:2302.05973 (2023). https://arxiv.org/abs/2302.05973

teaching

Direct Reading Program(DRP)

Mentoring, University of Texas at Austin, Math Department, 2018

Mentor for DRP on topics including ‘Support Vector Machine, Stochastic Calculus for Finance and Discrete Fourier and Wavelet Transforms, Quantum Mechanics, Numerical Partial Differential Equations’.

Teaching Assistant

Undergraduate and Graduate course, University of Texas at Austin, Math Department, 2018

Teaching Assistant for undergraduate and graduate courses including ‘M408C (Calculus 1), M408D (Calculus 2), M427J (Differential Equation and Linear Algebra), M381D (Graduate Complex Analysis), M387C (Graduate Numerical Analysis 1), M328K (Introduction to Number Theory), M383C(Graduate Methods of Applied Mathematics 1)’