Organizers: Albert Kunhui Luan (kunhui.luan@sc.edu), Qiyu Wu (QIYU@email.sc.edu), Jesse Singh (jasdeep@email.sc.edu), Haonan Zhang (haonanz@mailbox.sc.edu)
This is a student-led seminar that focuses on applications of harmonic analysis to PDEs and mathematical fluid dynamics, including the analysis of singular integrals of the Calderon–Zygmund type, fractional Laplacians, various interpolation results, and Littlewood–Paley localization. The discussion will be based on selected papers and references listed below.
Papers
- Blow-up and regularity for the fractal Burgers equation
- The fractional p-Laplacian evolution equation in ℝN in the sublinear case
Literature References
- The Mathematical Analysis of the Incompressible Euler and Navier–Stokes Equations
- Classical and Multilinear Harmonic Analysis, Vol. 1
- Hitchhikerʼs Guide to the Fractional Sobolev Spaces
Meeting Time & Location: Every Wednesday, 10:30 a.m. – 11:30 a.m., LC 346
Everyone is welcome to join! There are plenty of opportunities to read parts of the papers and present results.
2025 - 2026 Academic Year
Date: September 10, 2025
Topic: Viscous and inviscid Burgers equation; regularity vs. blow-up, Cole–Hopf transformation, and introduction to fractional Laplacian operator.
Date: September 18, 2025
Speaker: Albert Kunhui Luan
Topic: Equivalent definitions of the fractional Laplacian; removing its singularity at the origin. Realizing the fractional Laplacian as a pseudo-differential operator of the multiplier \( |\xi|^{2s} \).
Date: 09/25/2025
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Topic: Regularizing effect of fractional Laplacian and the global regularity of fractional Burgers equation with subcritical index \(s > \tfrac{1}{2}\).
Abstract: Examine the regularizing effects of the fractional Laplacian through its Fourier representation, with the aim of establishing global regularity for the fractional Burgers equation in the subcritical regime \(s > \tfrac{1}{2}\). In particular, we investigate the \(L^\infty\) maximum principle control.